My opening repertoire so far includes pretty much exclusively the French Defense as Black against e4, and that is about it. What does this indicate about what type of player I am?
Also what would be some good suggestions to play as White? I tried the King's Gambit a lot but it either is just not my style or maybe it requires more study than I have put in to get decent results with it.
I agree that 1.d4 and Queen's gambit are the best option from what you told about yourself.
In French, it's typical to have d4-d5 pawns in the center, a bad black white-squared bishop, space advantage for White and a plan based on c7-c5 for black. These points are also common in the Queen's gambit declined. I write from my own experience of playing these openings.
You might also search through a database and see which grandmaster's style fits you well, and copy their opening repertoire. Good luck!
Do you like the french defense?
If you prefer more halfclosed/closed positions (which often arise from the french defense), then i would suggest to play queens gambit because it focus more on positional play rather than attacking right out of the opening like king's gambit.
But besides that, you should know some opening theory regardless which opening you play.
Basically (really):
If you have an attacking style, with a preference for combinations with a lot of space to move your pieces, then you should go for open games (e4, as a first pawn move, can generally lead yout to that kind of positions, example : the Sicilian) King's gambit (1.e4 e5 2.f4), for instance, is maybe not the best way to appreciate that (too cutting edge and still dangerous for white), instead you could try (1.e4 e5 2.d4) which can put you very quickly on the battlefield.
On the contrary, if you like slow, closed, positional games with long piece maneuvres, then you should go for d4, that generally leads to closed games (exp: Queen's gambit, semi-slav defence) !
Anyway, the better way for you to know waht really suits you is to try as many different openings as you can, until you get it !
If you are an attacking player who likes to develop quicker, I would recommend the Danish Gambit or Evan's Gambit. Even if declined, white can get a pretty decent advantage in development and can attack the weak pawn of f7 on the Black board. However, if you are a solid player, I reccomend the Italian Game or Reti's Opening. I owe a lot of my success in the intermediate level to these openings as white. As black, I nearly always play the Sicilian to e4 and the King's Indian against d4. The few times I do see c4, I play e4 and try to get a reverse Sicilian position.
I read your question. To answer it, it would be necessary that you provide more détails on your "personal" situation. I mean, what are your strength (elo?), your age, your experience, your motivation (time to learn), your aspirations? How long have you been playing, are you a newcomer to chess? What did you play until now with white (with what results), what is your preference: slow, positional, tactical play, both, manoeuvering, middlegame, opening rapidity of motion/development, do you like open or closed games. For instance, but just for instance, if you like slow positional, almost contactless play, you can adopt (with modifications to be made for opening developments since publication) the repertoire proposed by Emms in Attacking with e4 (an "old" but very good book). However, to continue with my example, if you are young and you want to learn and to make progress, there is a precondition for progressing, it is mastering tactical basic schèmes (all of them) and more advanced ideas, to be acquainted with tactics of everyday (by this i don't mean basic themes, as someone else understood here, but the subtle interweaving of themes and foremost the concrete calculation necessary to accomplish plans), and so on. Which kind of French positions do you like and predominantly, what don't you like as regards pawn structure types? You can furthermore take into account the frequency of types of positions played in the opening you choose. I mean if you play the Rubinstein French and you like a specific variation and dislike other variations and the statistics say the variations you don't like represent 80% of the played French Rubinstein games, then you should make a more pragmatic choice to increase your chance of getting a convenient (to you) position type. This does, as is quite evident, not mean that you can avoid all unwished variations. Furthermore, you should be aware, should you wish to play a reversed French, that the difference between the french and the reversed french is the fundamental nature of it. The french is concrete and immediate whereas the reversed french, while offering no special advantage, postpones the concrete and direct confrontation in the late middlegame as a result of the absence of contact.
If you're style is defensive, d4 would be nice, it avoids Sicilian which can be very aggressive with opposite castling and most lines are passive with not much attacking potential.
If you are playing queen's gambit, beware of Albin countergambit as the game can be highly aggressive.
If you're opponent plays (Old)Benoni Defence, I recommend 1. d4 c5 2. d5 Nf6 3. c4
1. d4 c5 2. d5 Nf6 3. c4 e5 will be a very passive game
1. d4 c5 2. d5 Nf6 3. c4 e6 might be slightly aggressive on the queenside, but you could should be able to prevent any potential successful attacks
As for Benko gambit, I recommend
1. d4 c5 2. d5 Nf6 3. c4 b5 4. cxb5 a6 5: b6/e3
For b6, the queen would take and white will attempt a queenside attack
For e3, you will be the one attacking, however, it should be easy as white's position is quite locked up by the pawns
Now if you're opponent plays dutch, it is defiantly an attacking game. Dutch is normally rare as it does nothing for black development, it only starts an early kingside pawnstorm but weakens their kingside, so you'll just have to defend the kingside or attack with your opponent.
I'll also recommend you to learn the London system which is a highly passive game and normally is a draw. If a draw is fine for you, you would want to look up the London system.
P. S. If you send some of your games, we could find openings that suit your style
