What parts do I need to replace for my bike?
Provided the bike has been kept dry in the intervening years then lube it up, possibly replace the tyres and you're good to go.  However I suspect the problem is your increased height, and to a lesser extent, weight. For your weight, the only things to consider are tyre pressure and maybe stiffening up the forks a little if that is possible.

However, to get the bike to suit your height will be tricky, a 17.5" frame is pretty small for a 6'1" person.  I'm 5'10" and my Specialized of the same size and era feels small after riding properly sized modern era bikes.  You'll probably require an extended seat tube and maybe a longer stem.  The cranks are likely to be 175mm anyway, so you'll have limited choices for anything longer.  But I think it'll end up being less than ideal, the highly extended seat post in particular being a weakness.

I think if you were to try a modern bike for size you'd find the long frame / short stem geometry and 27.5" wheels to your liking.  You're going to need to go to a bike shop to get the seat post and any other bits anyway, so there's no harm in trying.  If you don't like the feel of the new bikes, do up your old one so it kinda fits.  Otherwise shine it up and make back some money to help fund your new purchase.