Look at this diagram:
And here's the link to the site when I found it:
http://mauiwater.org/fixleak.html(Yep, Maui Water Works :) ... as in Hawaii .... but the info is very valid)
The info on the site is for a faucet that is part of a larger faucet assembly. Which is just like your sink, but a bit different from your bathtub. That said, the innards are pretty much the same. These are called 'stem-type' faucets. The stem is identified in the diagram (the big brass colored piece in the lower center). The failed part is the rubber washer at the lower end of the stem.
These instructions, like so many others, start out saying to shut off the water supply to the faucet. For the tub, you may well not have such a shut-off. Find the main water supply to your whole house and shut that off. I have no idea where that might be in your house. In mine, it is in the basement, right where the water enters the house from the street. In any case, shut off this valve. Now open the leaky faucet and allow it to run until it stops flowing.
Now you're ready to work on the tub. I'd bet your valve is assembled exactly like the one in the diagram. Take it all apart. Start by removing the screw in the center of the faucet handle. You may have some trouble with the faucet handles; they tend to corrode and become one with the stem. Normally, a good, sharp yank will get it off. If not, you'll need one of these - a faucet handle puller.
This works by putting the center pin into the screw hole, putting the hooks under the edges of the handle, and tightening the 'T' handle. This will gently pull off the handle. This tool, should you need it, is probably in the $5 range.
Now get everything apart. The instructions on that Maui site, linked above, are pretty much what needs to be done.
Now ... do NOT remove the rubber washer at the bottom. Take the entire thing, washer still in place, to a hardware store (the local Ace or True Value or independent store is, for this next step, waaaaaaaaaaaay better than a big box like Lowes or Home Depot). have the hardware store guy remove and replace the washer. The washer is probably all deformed and worn and buggered up. If he has the whole stem in his hand, he can match the stem to the washers he'll have in stock. Also, if there happens to be an O-ring (not shown in the diagram) on your particular stem, have him replace that too, no matter how good it looks.
Now you're ready to go home and reinstall it. Reinstalling is simply the reverse of the removal.
If the faucet handle was hard to remove, consider getting a new one. They're pretty cheap. The Ace guy will have them, too.
Also, if there's a lot of white corrosion on your brass stem, before you go to the hardware store, you may want to soak it in some lime scale remover (CLR is a good, commonly available brand - try the grocery store). This step is completely optional. I just do it as a matter of routine. I dunno why .... just 'because'.
Now, you also lucked out in that your shower diverter is in the spigot. It is the thing you pull up to allow the shower to run. Some tubs have it as a third valve. Not difficult, it is a bit more complicated than the water valve. But you can forget it. You have one that needs no maintenance.
Okay .... now that sink. It is pretty much the same process as for the tub. If you have the water off for the tub, that's the perfect time to do the sink, too.
On the other hand, if, for some reason, you decide to do the sink another time, you may feel it is better to shut off the water to the sink using the shut off valves that are almost certainly located below the sink. Don't!
Shut off the water at the main, like you did for the tub. The shut-off valve below the sink is a cheap one. They are notorious at not shutting off (turning) easily, and then leaking when you try to reopen them. Avoid the hassle and use the main valve.
Anyway, that's all there is to it.
Let me know if any of this is unclear or if you have any other questions.
Bon chance! :hi: