Focusing on the 3. g3 line, you have the Advance Variation of the Réti. There is an extensive discussion of this line available here, an offshoot of this page on Understanding the Réti.
These are the main lines listed there:
Main line:
a. 3...Nc6 4. Bg2 e5 5. d3 Nf6 6. O-O
b. 3...Nc6 4. Bg2 e5 5.O-O
c. 3...c5 4. Bg2 Nc6 5. d3
d. 3...g6 4. Bg2 Bg7
Critical line:
e. 3...Nc6 4. Bg2 g6 5. d3 Bg7 6. O-O
The reason for the popularity of 3. g3 is that by advancing to d4, black has weakened their hold on the center, opening the way for control of the center diagonal with a fianchettoed bishop. As noted by Bad_Bishop this is a reverse Benoni, with the crucial difference being that you have an extra tempo to play with. Alexander Delchev has a good book on the Réti, with 28 pages devoted to the reverse Benoni (The Modern Réti). As Delchev notes, you're likely going to want to make the most of your extra tempo and play a very sharp game. Unfortunately, this counts against 3. g3 assuming Black counters with 3. ...Nc6!. If, however, black plays 3. ...c5 then you can counter with a strategy similar to the Benko (likewise for 3. ...g6). I'll start with an overview of the kingside fianchetto, and then delve into specifics.
General Strategy for a Kingside Fianchetto
The purpose of 3. g3 ... 4. Bg2 is to exert control on the central h1-a8 diagonal with the fianchettoed bishop. This bishop will become the lynchpin of your King's defense, since any move of a pawn in front of the king opens lines of attack, as well as a great tool for indirect control of the center in the hypermodern style. This bishop is very valuable and should not be traded without significant advantage as compensation. The e2 pawn is usually important to keep in place to help protect f3 and avoid pesky attacks from a knight (but this rule of thumb has many exceptions). A common strategy for trading off bishops is for black to double their queen behind their bishop on the c8-h3 diagonal and then push Bh3, attempting to pin the fianchettoed bishop to the rook on f1 (after castling) and trade with Bxg2. In the event of a queenside castle by black, you could wind up with a pawn attack on the h-file, aiming to break up the pawn structure of your fianchetto (this is the common line in the Yugoslav attack in the Sicilian Dragon). Since you're playing white here and already have a slight developmental lead, however, this strikes me as too slow to be practical.
You typically want to avoid a closed position to exploit the full power of your bishop's influence on the central diagonal. Usually you will attack queenside, with a pawn storm in slow games and careful maneuvering of minor pieces in quicker games. Given this, your pawn chain will generally extend from e2 to c4, giving you a structure similar to what you would find in the English, Reverse Dragon variation. Your big obstacle here will be how to open a file for your queenside rook, given that trading cxd5 is no longer an option. For this reason b4, leading to something like the Benko gambit, will be a popular choice in most of the lines following 3. g3.
Black Plays 3. ...c5
[FEN " "]
1. Nf3 d5 2. c4 d4 3. g3 c5
With 3. ...c5 (as well as 3. ...g6) you win up in the reversed modern Benoni. This move serves two main purposes: it protects the pawn on d4 and allows development of the knight with Kc6 without blocking in the pawn on c7. You'll notice that black doesn't have a lot of control over the light squares -- this is something that will work to your advantage.
You want to stick to your plan of development play Bg2 and d3, getting the pawn structure I mentioned above. The main line above makes these moves immediately, with 4. Bg2 and 5. d3. Going this route will generally lead to a slower game where you launch a pawn storm on the queenside.
The more aggressive option is the reverse Benko Gambit given by 4. b4, preparing the way to open a file for your queenside rook. This is usually accepted with 4...cxb4. With 5. a3 you force Black's hand, and will usually be met with 5. ...bxa3 6. Bxa3.
[FEN " "]
1. Nf3 d5 2. c4 d4 3. g3 c5 4. b4 cxb4 5. a3 bxa3 6. Bxa3
At this point you should see that you've gotten a considerable amount for that pawn. You have your rook on an open file, with another open file right next door on b, and you have developed your queenside bishop to possibly assist in a queenside attack (but likely not).
Black's strategy at this point is usually going to involve 6. ...Nc6 followed by e5, at which point you'll be forced to trade bishops on f8. Your plan is going to involve finally completing the fianchetto with Bg2 and following up with d3 for the e2-c4 pawn chain -- extending towards your point of attack. See the linked page for more discussion.
Black Plays 3. ...g6
[FEN " "]
1. Nf3 d5 2. c4 d4 3. g3 g6
Like 3. ...c5 this usually winds up in a variant of the reversed Modern Benoni, but a bit slower for Black. Given how sharp that defense is, the extra tempo can be exploited to great advantage. You're going to likely want to pursue a similar strategy involving a Benko-like push to b4, assuming that black eventually plays c5. Otherwise, it will likely transpose into the critical line (e) -- though Delchev disagrees that (e) is the line to worry about.
Black Plays 3. ...Nc6
[FEN " "]
1. Nf3 d5 2. c4 d4 3. g3 Nc6
This is usually considered Black's best line. Your plan will stay largely the same. You'll follow with Bg2 and d3, leading to a queenside attack. I typically prefer to hold off on castling, so would probably opt for the sequence in the main line (a). The point of 4. ...e5 is obviously the same as the move of the pawn to c5 in the other variation -- to protect the pawn on d4. You might be tempted to attack with f4, but I'd (generally) avoid weakening your fianchetto with such a move. The central diagonal is still open for your fianchettoed bishop, but you'll have to pay attention to the c pawn since c6 is still on the table once the knight is moved -- weakening your bishop's control of the diagonal. Better to simply follow with 5. d3 or 5. O-O (with the move you don't make coming shortly after).
Suppose the sequence goes as follows: 4. Bg2 e5 5. O-O Nf6 6. d3. The move Delchev considers best for black here is 6. ...a5!, which pre-empts the Benko-like push 7. b4.
[FEN " "]
1. Nf3 d5 2. c4 d4 3. g3 Nc6 4. Bg2 e5 5. O-O Nf6 6. d3 a5!
This sequence is actually what leads Delchev to reject 3. g3 as being too slow (he comes to a similar conclusion regarding 3. e3 which is also best responded to with 3. ...Nc6 and popularly continued with 4. exd4 Nxd4 5. Nxd4 Qxd4 6. d3 c6! which tends to cramp White's position and restrict its knight).
This line is one case where the e3 pawn seems advisable. See this game between Kujawski and Kiolbasa to get an idea of how things might develop. The basic idea will be to follow with 7. e3, which is best answered by 7. ...dxe3! since allowing White to initiate the trade with something like 7. ...Be7 8. exd4 exd4 9. Na3 O-O 10. Nb5 aids your development by giving you the active knight.
[FEN " "]
1. Nf3 d5 2. c4 d4 3. g3 Nc6 4. Bg2 e5 5. O-O Nf6 6. d3 a5! 7. e3 dxe3 8. exd4 exd4 9. Na3 O-O 10. Nb5
Now, the Critical Line according to the linked page is 3. ...Nc6 4. Bg2 g6 5. d3 Bg7 6. O-O.
[FEN " "]
1. Nf3 d5 2. c4 d4 3. g3 Nc6 4. Bg2 g6 5. d3 Bg7 6. O-O
Delchev actually doesn't consider this, having concluded that the previous sequence involving 6. ...a5! as decisive against 3. g3. From my analysis, I think Black's best strategy here is to simply follow normal development. The pawn on d4 is awkwardly placed since it blocks the fianchettoed bishop. So, I'd figure this would continue with 6. ...Nf6 7. b4, preparing to pseudo-sac the b-pawn in order to open a file for the rook and establish an attack.
[FEN " "]
1. Nf3 d5 2. c4 d4 3. g3 Nc6 4. Bg2 g6 5. d3 Bg7 6. O-O Nf6 7. b4
If 7. ...Nxb4 then 8. Qa4+ Nc6 9. Ne5 putting a lot of pressure on c6.
[FEN " "]
1. Nf3 d5 2. c4 d4 3. g3 Nc6 4. Bg2 g6 5. d3 Bg7 6. O-O Nf6 7. b4 Nxb4 8. Qa4+ Nc6 9. Ne5
Black can go with either 9. ...O-O or 9. ...Bd7. Either way you capture 10. Nxc6 and eventually get Black's b-pawn or a nice lead in development and good initiative. For 9. ...O-O we have
[FEN " "]
1. Nf3 d5 2. c4 d4 3. g3 Nc6 4. Bg2 g6 5. d3 Bg7 6. O-O Nf6 7. b4 Nxb4 8. Qa4+ Nc6 9. Ne5 O-O 10. Nxc6 bxc6 11. Qxc6
Whereas for 9. ...Bd7 there are two branches,
[FEN " "]
1. Nf3 d5 2. c4 d4 3. g3 Nc6 4. Bg2 g6 5. d3 Bg7 6. O-O Nf6 7. b4 Nxb4 8. Qa4+ Nc6 9. Ne5 Bd7 10. Nxc6 Bxc6 11. Bxc6 bxc6 12. Qxc6
and
[FEN " "]
1. Nf3 d5 2. c4 d4 3. g3 Nc6 4. Bg2 g6 5. d3 Bg7 6. O-O Nf6 7. b4 Nxb4 8. Qa4+ Nc6 9. Ne5 Bd7 10. Nxc6 bxc6
If you face the latter, I'd hold off on taking the pawn and develop your knight and rook with Nd2, Nb3, and Rb1.
Why does Delchev suggest 3. b4?
Though my main focus has been exploring 3. g3 and trying to provide some insights into the dynamics of play with a kingside fianchetto, Delchev recommends 3. b4 as the best move.
[FEN " "]
1. Nf3 d5 2. c4 d4 3. b4!?
His reasoning is that it offers you the sharpest play and makes the most of the extra tempo. You essentially jump right into a Benko-like gambit and exert maximum pressure. Optimal play continues 3. ...f6! (as you note) then 4. e3 e5! where it gets tricky. You have to be bold. VERY bold (not for the faint of heart). 5. c5 positions White for another gambit and is best answered by 5. ...a5!.
[FEN " "]
1. Nf3 d5 2. c4 d4 3. b4!? f6! 4. e3 e5! 5. c5 a5!
From this point, the main lines break into 6. Nxe5, 6. Bc4, 6. Bb5+.
The move 6. Nxe5 leads to a forced draw after 6. ...fxe5 7. Qh5+ Kd7 8. Qf5+ Ke8 9. Qh5+:
[FEN " "]
1. Nf3 d5 2. c4 d4 3. b4!? f6! 4. e3 e5! 5. c5 a5! 6. Nxe5 fxe5 7. Qh5+ Kd7 8. Qf5+ Ke8 9. Qh5+
With 6. Bc4 you face 6. ...axb4! then play naturally continues 7. exd4 exd4 8. O-O Bxc5 and now Black has two pawns and more space.
[FEN " "]
1. Nf3 d5 2. c4 d4 3. b4!? f6! 4. e3 e5! 5. c5 a5! 6. Bc4 axb4! 7. exd4 exd4 8. O-O Bxc5
It seems the line following 6. Bb5+ is the key. It leads to very sharp play, and while not necessarily to White's advantage, the knowledge imbalance should tip the scales in White's favor. The theory ends with 6. ...c6! 7. Bc4 Ne7!? 8. O-O!? axb4 9. Bb2, at which point Delchev concludes that White has compensation for the pawn and the attack.
[FEN " "]
1. Nf3 d5 2. c4 d4 3. b4!? f6! 4. e3 e5! 5. c5 a5! 6. Bb5+ c6! 7. Bc4 Ne7!? 8. O-O!? axb4 9. Bb2
The critical line continues 9. ...dxe3 10. fxe3 Nf5 11. Qe2 Bxc5 12. d4 Ba7!:
[FEN " "]
1. Nf3 d5 2. c4 d4 3. b4!? f6! 4. e3 e5! 5. c5 a5! 6. Bb5+ c6! 7. Bc4 Ne7!? 8. O-O!? axb4 9. Bb2 dxe3 10. fxe3 Nf5 11. Qe2 Bxc5 12. d4 Ba7!
At this point the three branches considered by Delchev are 13. a3?! e4, 13. dxe5 Nd7!, and 13. Nbd2 Qe7 -- none of which are simple or clear cut. Engines can hold the line for Black, but things are complicated to say the least. From what I can tell, the idea is a continuation of the theme of this variation: make things sharp, make things complicated, and hope you know more about the opportunities and pitfalls of the variation than your opponent. For what it's worth, I play the Sicilian Dragon with the same strategy in mind. If my opponent knows the Yugoslav attack -- and they usually do if they're any good -- I believe they have a slight edge on paper. But the opening is so sharp and bookish (I believe the book line is settled for something like 22 moves) that unless I'm against another Dragon specialist they can rarely avoid some mistake that I know how to exploit.
The Bottom Line
The line with 3. e3 is safe and slow, perhaps too slow (though I haven't discussed it in detail). At any rate, it isn't my style. 3. g3 has potential if you're able to exploit some of the gambit opportunities, but could be turned into a slower game in the 3. ...Nc6 line. 3. b4 is very aggressive and sharp, but an interesting option if that's your thing. I'd probably go with 3. g3 since I'm more familiar with that sort of pawn structure and I'm a big fan of keeping the pawn structure similar across my various openings/defenses. If, however, you're not comfortable playing with fianchettoed kingside bishops and aggressive gambits then it might be a bit much for you to jump right into.
