Since you're playing live and not in a studio, it wouldn't be wise to start out with a load of vintage analog synths. Or even one for that matter. Trying to keep it a vintage analog synth in good playing condition is a pain -- even for someone with a lot of analog experience.
If you don't believe me, check out the Analog Heaven mailing list. Almost 50% of the posts seem to be people getting or giving help for maintenance and repair issues on vintage synths.
A good sounding replacement for the Mini can take a couple of routes:
First, for keyboard, there's
an Alesis Ion. It's digital, but it mimicks a lot of the old 70's synths (Moog, Arp, Oberheim) very well and is used by a lot of people to sub for a mini. On top of that its not that expensive, there's a ton of pre-made patches for it, there's a boat load of help available for it on the net and it's a great starter machine. YOu can start simple and get pretty insane with all the modulation routings and other goodies under the hood.
Another alternative if you want to remain pure to the analog realm is
the Studio Electronics SE-1(x). It's a rack machine but 2nd hand they are affordable.
Another rack -- and if it's genuine Moog you're after, there's always the Voyager rack, called the Rack Mount Edition (or RME).
My point is, that unless you've like got an insane amount of money to burn, start cheap and work your way up.
An Ion and a Rompler (someone mentioned the XP-30. Good choice with the vintage sounds), will do just wonderful to get you started. Then, after your out or your Mom's garage and are out giging you'll be in a better position to know for yourself what you want in your keyboard rig.
And there's nothing wrong with starting in your Mom's garage. I did, and I'm sure 99% of everyone else here did too!
-bruce