You missed out a lot of important shit in your post, so you should pay attention to what I post here too.
Alright, so Big Band in a nutshell is a heaps defensive/poking character in a super offense heavy game.
How do I keep my opponent at bay with big band?
The most potent and subtle way to prevent your opponents from approaching you is to make it seem like a really bad idea to approach you - Big Band is good at this, because he can poke half-screen. Setting up your spacing game, and being able to properly threaten areas of the screen is the key way to keeping opponents hesitant, and if they're hesitant, you can slowly close in and gain space and put them in the corner.
Of course, what Skymin said about "put them in the corner first" is true - A good offense is the best defense in skullgirls, if they have to deal with your shit, you don't have to deal with their shit.
What is the best anti air move?
Anti-airing with Big Band is situational and difficult from the ground. If you want to keep your air space safe reliably, prevent approaches by utilising air-to-airs such as Big Band's godlike j.LK bicycle wheel thing, since it's got a super good hitbox, or his air qcb HK since it reaches real far (but not as far as the graphic would indicate).
What is the best ground defense move?
On the ground, c.MK and s.MP actually have pretty good anti-air hitboxes but they have long recovery animations, espcially s.MP, and if you use s.MP you could accidentally go into play-the-trumped mode which means you die instantly. s.LP is good against instant airdashers but that's about it.
His anti-air A-Trains are good but are more of an anti-jump offensive option since they are unblockable if the opponent is ascending but can be blocked if they're descending, they have some utility vs single-hit jumpins since they'll have armour and eat the hit but that's about it.
Last but not least, parry into LP Beat Extend. This is by far his best anti-air option since it negates the hit, Beat Extend will blow through followup hits, it's too fast for your opponent to react with super, and it will hit no matter what height your opponent is at. Yes, the scaling is bad so your followup combo does poor damage but you could end in a slide knockdown with Hard rush punch or anti-air grab to set up a mixup, or do a reset of some kind. This option difficult though, since parrying is a large commitment and requires both prediction and reaction and if you are waiting for the parry, then you aren't putting moves on the screen for your opponent to respect.
Parrying is difficult but highly rewarding when you get it down. It's highly recommended you use a character that doesn't have parries in addition to Big Band so that you don't over-rely on it.
Now, onto offense. Offense with big band is tricky but potent.
How do i safely approach my enemy, especially keepaway characters like Peacock?
There's no magic bullet here, at neutral you're going to need lots and lots of patience, and/or make some very specific reads. Big Band really shines if you can make some real smart reads to quickly get momentum, and his mixup game post-knockdown is freakin' sweet.
Regarding patience, space out the opponent, make them respect you, and slowly eke your way forwards with double jumps and blocking, while keeping their approach out with j.LK should be the basis of your game. If you just relied on that, however, you will get your shit kicked in since it's not that potent by itself.
Regarding All versions of brass if you think they're going to move forward, L if you need a quick one, H if you know you need to plow through some normals/projectiles, M for something inbetween.
A-Train is good for opponents that like to jump, airdash, and doublejump so if you sniff out those movement options you will get a knockdown.
HOWEVER.... There is a way to make all of these unsafe, risky reads quite safe indeed. You get an assist. For instance, you can just pick up Double, use her MK Hornet Bomber assist, and if you call it just before doing a ridiculously unsafe lunge punch or A-Train, the butt will cover you and you'll be safe anyway. This makes Big Band stupidly strong at mid range.
What are the best moves to start a combo?
The most reliable situations Big Band can start a full combo from is from one of his SLIDE KNOCKDOWNS. You can get a slide knockdown if you have preserved your purple bounce, and you hit your opponent with either a H Brass, or H A-Train. This means they can't ground tech, and they have to get up at the same time every time.
j.LK chained to j.MK is real good when you're ontop of someone, since it hits people stand blocking, and if someone is waking up from a slide knockdown they are in standing state EVEN IF THEY ARE CROUCH BLOCKING. This is good pressure as well since you'll be plus on block and they'll have to pushblock you out.
c.LK chained into c.MP c.MP is also real good when you're next to someone, since it has to be blocke low, and if they start to expect your j.LK they'll block high, and you'll catch them low. Also, if your opponent is jumping on wakeup, then c.LK will keep them in pre-jump, they get hit, and it lets you continue into combo.
Bonus, if you have an assist(like double's MK butt!), then on their wakeup from slide knockdown, call the assist, do instant j.MK. It hits high, and they'll continue to get hit by the assist and you can do full combo.
Big Band can also start combos with his ground throws, and air throws. Do them if you think they'll block on the ground, or be blocking in the air. You can even do air throws aggressively at neutral!
What most basic and simple combos involving special moves should i learn as a newbie player?
I'll give you the combos I use, since they're relatively simple, not really optimised, but they get the job done.
Full combo Midscreen:
whatever starter(Probably c.LK c.MP c.MP), s.HP, j.MP j.HP j.HK, otg c.MK xx H Beat Extend, s.LP s.MK, j.LP j.LK j.MK c.LK c.MK s.HK xx H A-Train, Super.
Full Combo Corner:
whatever starter(Probably c.LK c.MP c.MP), s.HP, j.MP j.HP xx qcb HK, s.HK xx M Brass, c.LK s.MK j.LP j.LK j.MK, s.LP LP c.LK c.MK s.HK xx H A-Train, Super.
However, you really shouldn't end your combos with meter usage if they're not going to guarantee the kill. In that case, just do
whatever starter(Probably c.LK c.MP c.MP), s.HP, j.MP j.HP xx qcb HK, H Brass.
This will get you the slide knockdown. As I mentioned before, Slide knockdown is godlike. Use it to your advantage. It's Big Band's most consistent way to mix his opponents up.
That's all I can think of atm. Use what I wrote here as the basis for your game, strongly consider picking up Double for her assist, and explore Big Band further in your own time.
