The key thing is pacing yourself through the battles at the beginning when you don't have many stamina (green) bars. You can't just keep going in wailing away because you'll eventually get hit and comboed to hell over and over. If you can, always wait until you have at least 2 bars of stamina for a vanish before engaging the enemy. You can pace yourself by finishing a combo, launching the enemy, and then not pursuing the enemy to give yourself some time for your stamina to recharge rather than dashing in (which uses stamina) to continue the fight.
Another key thing is using the back attack to interrupt an enemy's vanish. This prevents you from getting into a stamina draining match where you combo the enemy, he vanishes behind you and starts comboing you, you vanish and repeat until one of you runs out of stamina. Successfully pulling off a back attack puts you ahead in the stamina war: in fact, that's what melee battles are all about where the person who runs out of stamina will probably lose or take heavy damage.
Against multiple opponents, you need to be especially patient and pick your moments to engage since it's very easy to get interrupted. I highly recommend using throws or instant launch attacks (I can't remember the buttons at this point) to basically CC (crowd control) one of the opponents out of the fight momentarily to give you a window to combo the other enemy uninterrupted.
To optimize your damage, you need to learn what the melee combos do (launch at the end? What direction does it launch? Sets up for another combo? Does it include a teleport which might help with positioning?) and make up your own combinations which might include melee combos followed with supers or even ultimates. For example, you can easily combine a melee combo that launches with Kamehameha, Galick Gun, or the ultimate Full Power Energy Blast Volley (this ultimate only works well if it's part of a melee combo. Without a combo, it fires the blasts in random directions but, if following a combo, all of the blasts home in on your target. On that note, I should mention that every move has its eccentricities which you need to figure out).
This is a really good video that shows the basics and starts introducing more advanced combos:
https://www.youtube.com/watch?v=GctOnMdKYDY
For managing 1 vs. many fights, here's a video that I recorded that involves 1 vs. 2 and 1 vs. 3 fights:
https://www.youtube.com/watch?v=QWKQehpajjA
And here's video where I was trying out a combo that I had put together which strings together a melee combo and two supers which you can chain together indefinitely if the enemy doesn't vanish or escape move out of it. I get to fully pull it off on Piccolo who basically loses 3/4 of his life to it:
https://www.youtube.com/watch?v=3vfYIK9JtyU