I have never done this. Ever. I am not very skilled at any of this but am willing to try.
I have a coffee table. It has green legs that I want to paint a different colour (they used to be wood, have been painted over - with some drip marks that I don't love), and a wood top that I can leave as is or stain?
Do I have to/should I strip the old paint off? Sand it? Leave it and paint over? This is really my biggest question!
I would strip off the old paint, sand it down as best as you can, and then use multiple thin coats of the new paint color of your choice. It'll take longer than just painting over the old, green paint, but it'll look much nicer and last longer. Once your final coat of paint has completely dried, I'd throw on a coat of poly to protect it. Wait a few days for it to cure before you put anything on top of the table, just to be safe. Hope that helps! Otherwise, there are tons of articles and tutorials online, too!
You will need to strip and sand the table legs. Stripping the table legs will get rid of those paint drips. Sanding the legs will prepare the wood to accept new paint. After you sand, wipe the legs down with a wet cloth and then a tack cloth. This will get rid of all the little bits of sanding. The legs will look much better. Next step is to paint. Paint with the grain (lines in the wood). If the lines are vertical, paint vertical brush strokes. Allow the paint to dry 6 to 8 hours. Then apply a second coat and possibly a third with 6 to 8 hours drying time in between. Paint fumes give me a headache so I paint in the wide open garage with door open. Poly is also a good idea too.
How old is the paint? Any chance you're dealing with lead paint? If so, you'll need to be a little more careful, though there's probably not much paint on the legs. I use plastic tarps under the object I'm stripping and I wear a disposable coverall from HD and a lead-safe respirator when I'm stripping. I don't sand on it until all the paint is gone either so that there's no lead dust floating around. I double bag everything at the end.
As far as the method, I've used Jasco (burns like HELL if you get it on you, but it's quick working), Citrus stripper (easier on the skin if you get it on you, doesn't smell bad, takes more applications for me than Jasco), Peel Away (takes a day or two to cure, learning curve, less effort, very expensive) and a heat gun (no chemicals, possibility of burning wood). All have their pros and cons, and I switch back and forth during a big job based on how the paint is reacting. If the paint is really thick and old and on a flat surface, I find the heat gun to be the best. If it's thin, the chemical methods work better for me. For large areas, I like Peel Away.