the cb350 has a ton of support. you can race it with the uscra, wera, and ahrma. its a very very generic bike probably similar to the sv, or ex500 now. but you can make them go very fast. and there are plenty of big guys riding them.
as for getting on the track, you dont need a race license to do a track day. but youll most likely be held in slow class with minimal passing etc. unless its just a free for all. go to the hawk school get your racing license. its as cheap, and more worthwhile. then you can do track days as a racer and not a squid, plus you can race.
get the 350 desmo, and hang onto it. youll want to race that after you get bored learning with the honda.
go to the racing school, then you can join any club and hold a license with that club. you just need to prove youve done the school. the uscra is by far the easiest and most laid back for getting your papers. and we're nice guys to boot. plus being in ny, there are alot more people local to you then say wera guys but there is alot of cross over from club to club. most clubs to get you onto the track cost about 100 a year. (and yes im including the uscra since you have to be an ama member also.) you get a news letter (except for wera) and a sticker pack, rulebook, etc...) youll need a new decent helmet, some good boots, quality leathers, and gloves. for the school, your bike doesnt have to be the full on race prep. but the closer it is, the better off youll be. you'll hav to contact each track day club and see what they require. it maybe that for the hawk school, you dont need to do alot. definitely get some good tires. most people run avon race tires. worth it. and brakes. suspension is a great upgrade too. then youll need to source pipes, but branson is an awesome source for race bits. if you go over to www.race-uscra.com
there is a rule book and more info. you can also try wera.com, but its not as easy to figure out i dont think. also, go to the track. make sure its something you want to do. theres a bit of an investment to realize you get out there and its not for you. everyone should try it once though.
and take kens advice.