> what is the best tradeoff between learning some lines really deep vs learning many variations but not as deep?

There are several cut-offs you can use:

 1. Cut short the depth in a particular line when it becomes "calm". So, both sides have developed and there are no immediate tricky tactics.
 2. As a first cut on breadth, make sure you include critical lines which have tactics which you need to know to survive. Do this even if they are relatively unpopular. You don't want your first game in that variation to be a quick loss.
 3. For less critical lines cut short the breadth according to popularity, unless you particularly like an unpopular line. In general you get the most benefit by studying the most popular lines first because they are the ones you will face the most often. Use a large database to identify these and make a cut-off point in popularity below which you are not going to spend much time. Only study an unpopular line after you have faced it as part of your post-game analysis.
 4. Another useful breadth cut-off if you don't have a great deal of time to spend is to do so according to how much your choice limits your opponent's choice. Examples would be: As an e4 player with white against the French only study the Exchange Variation (even though it is less popular) because it allows *you* to limit your opponent's choice of moves and so drastically reduce the amount of learning you have to do. Same argument for playing the Exchange Variation against the Slav.