Welcome to the forum.
There are some big questions in that post.
Mechs: A lot depends on condition, but if the bike already has something like Shimano DX mechs, then there is no real reason to change them, unless you really want 9 or 10 speed which they may well struggle with.
Shifters: You can't really go wrong with thumbies, as a lot of old sti shifters will have gone all sticky by now. That's not a massive issue to fix, but thumbies are pretty bomb proof.
Seatpost and seat: Modern items are pretty good, although you can now get new Turbo and Flyte saddles, so it will still look retro.
Handlebars: Depends on the stem and your ride position, but unless you really need to have fixie type narrow bars, I'd be inclined to go in the opposite direction and get some flat wider bars 560 to 600mm wide. You get a good degree of control from them and the steering is a lot less twitchy.
Wheels: Light wheels make for a better ride, but watch those rims if you are going to be jumping on and off kerbs and the like.
That's my 2 bob's worth, I'm sure others will be along with their own ideas.
Most important thing....have fun.
"Try not to punch any clowns"