About a month ago, I planted a Manila Mango tree, then topped it. Now it's sending out many shoots out of nodules in different directions. I see about seven or more future branches coming out. How many should I keep?
If it were mine I would keep them all and let let grow out a bit, then you can take a look at it and see what shape its taking and "tip prune" them where you want.
Below is a Video on Tip Pruning...just in case you haven't already seen it...Good luck!
Here is a link that might be useful: Video: Pruning--Tipping Mango trees...
Be aware that it makes a difference where along a branch (or single stem) you tip a mango. If you tip just above a growth ring of buds (indicating where the last flush started) you will get a lot of side growth concentrated at that spot: brush-like.
If instead you tip/prune one or two nodes above or below such a ring then you will get branching from a few nodes (generally three) below the cut. These have the advantage that they are more spread out.