One of the best pieces of equipment you can use for keeping an eye on variable stars is a good pair of binoculars and/or a short focal length scope...like a small refractor. You use the binos to get a good look at a wide field and then zero in on the star with the scope. But even the brighter variables are quite amenable to observing with binos. Once you get a good eye in on discriminating the brightness differences in stars, you can then track any changes in brightness for many variables.
Ultimately though, you'll want a good scope with an EQ mount so you keep track of your star and a
photometer to measure any brightness variations. Quite a few variables can only be measured using sensitive equipment because their variations are so small and/or rapid. You'll also have the added benefit of being able to track transiting exoplanets as well, with the right gear.