First of all: Welcome. 10 speed bikes can be a great find. Sorry for the following wall of text.
You could do the full monty: upgrade everything. Too pricey, lots of tools needed, not good.
Better:
Replace the stem mounted shifters with bar end shifters (though I personally prefer down tube shifters, depends on what you're used to I guess). Make sure those are friction shifters (or can be set to that). Replace break levers (make sure they work with the current brakes), if it is really the brake levers that are broken, the hood caps can be replaced easily. Change the handle bar tape while you're at it. You might get those Aero brake levers where the cables are routed under the bar tape, no big price difference.
You need: replacement parts (levers), cables + housing + ferrules (both for brake and shifter), a good cable cutter, hex / allen wrenches (depends on what components you have), screwdrivers, an awl and a file (to file down the cut ends of the housing and open it carefully). Handlebar tape + electric tape (matching colour), scissors / knife. Also replace the brake pads while you're at it!
What you should also check for: chain wear, bearings (hubs, head set, bottom bracket), though you will need more tools for these: flat cone wrenches and tweezers for the hubs, big hex headset wrenches for the headset and some strange tool for the bottom bracket (depends) + crank pullers (depends).
I would go for the cone wrenches (a set is about 7 EUR = 10 USD?), get a small tub of marine grease and some long tweezers. You also need somebody to remove the freewheel. You could get the tool for that but in my opinion it makes no sense: You will end up with something more modern someday anyway and at least bike shops here charge nothing for removing the free wheel (if it is not problematic).
The bottom bracket I'd leave in its current state or give it to a local bike shop to open, regrease and set up. The head set: depends... if you have access to a workshop with big hex wrenches: do it, though I expect the bearings to be worn. Still, new grease is always good and you'll learn a lot.
Replacements down the road:
- wheels: steel frames can be forced (cold set) to accept slightly wider modern wheels.
- chain (probably soon / now)
- bottom bracket (if it is really worn)
Just be sure to think about all of the "nice to have" things. In the end you will probably have a nice bike to ride but you'll sink a lot of money into it. I know I did.
Thank you, this is helping me out tremendously! I have already had to replace the bottom bracket after a month of riding, it was completely shattered on the inside, and the rear wheel has a slight bend in it, so I was already looking into replacing it. The brake pads on it are brand new as well.
The brake levers are falling apart, when I brake with them, I can see the plastic splitting open, so it is only a matter of time before they completely fall apart. As for the shifters, I found some bar end shifters for about $75 on ebay, which seems to be the lowest I can find.
So my only question now is how do I go about installing them. I'm pretty confident I can replace the brake levers due to they seem fairly simplistic. Also, this is just due to me not knowing much about bikes yet, but why do I need to remove the freewheel?