Different corps have different priorities. There are some that want nothing but small and cloak-specialized ships in fleet. [/shrug] For the most part, you'll usually be allowed to fly whatever. (Unless you insist on flying a Bantam for EVERYTHING
Try to gather a few different ships geared for the different roles you may be needed for. (Very few Fleet Commanders will turn down an extra E-War pilot)
As for operations...again, depends on what the corp does. An Industry corp will do plenty of mining. An all-out combat pilot can still make a contribution to mining ops by running cover for the miners.
A pirate corp....if you get caught with a mining laser, the first words you utter must be "Look what I just looted from that carebear's wreck!", and then quickly sell said laser. (Unless you're running a mining laser as part of being bait)
Take a look around. Different corps will have different agendas and methods. Do a little research before applying to a corp. Talk with the members a bit and find out if you can stand to spend any time with them, or do the things they prefer to do.
For the complete newbie, there's always Eve University. They specialize in training the helpless newb into someone who's not a hazard to himself and those around him. Having Eve University in your resume is NOT a bad thing at all.