Tires are good upgrades, and aren't wasted since you keep the old ones as spares. You need spares.

I think wheels can be reasonable upgrades. Not that there's anything wrong with freewheels, but a freehub is better. The wheel can be lighter, stronger, sealed bearings. It makes a difference.

It's true that upgrading parts will cost you more in the long run than upgrading the whole bike. It's also true that any upgraded part is just as much better than the old one on a beater bike as it is on a high end bike. If it saves you 100 grams on the high end bike, it saves you the same 100 grams on your bike. Snappier shifting is snappier, whichever frame. Is it worth it in either case, who can say, but if you like to tinker and enjoy having spare parts around from the take-offs, it's not unreasonable to upgrade.