The key features to look for in a mounting is stability and smoothness of movement. You don’t want a wiggly, jiggly mount! You also don’t want a mount that moves in erratic fits and jerks. Tap the telescope’s tube when you’re looking through the eyepiece and see how long it takes the vibrations to damp down! If it takes more than 2-3 seconds, the mount’s too wiggly. Try moving the scope and feeling whether there’s any binding or sticking, or alternatively if a slight tap sends the tube spinning around like a top! Either situation is undesirable; the scope should move freely but not TOO freely (or it will be buffeted about by every light breeze), and it should stop moving the instant you stop nudging it or adjusting the controls (whichever the case may be). Think “smooth and controllable” when assessing the movement of the mounting.
There are two basic types of mountings: equatorial and alt-azimuth. Either can work well if it’s well-constructed.
An equatorial mount has one axis of movement tilted to match the tilt of the Earth’s axis; if you align the mount so that axis is pointing to toward the celestial pole, you can easily drive the other axis (either with a manual slow-motion control or with a small motor) so that an object will stay in the center of the field of view while you observe it. The downside of equatorial mountings is that they can be cumbersome and it takes some practice to learn how to set them up and use them properly, and they’re more expensive than an alt-azimuth mounting (for the same size telescope). But some people find the easy tracking they provide worth the added expense.
Alt-azimuth mountings, like the name implies, move up-down and right-left. To keep an object in the field of view with one of these mounts means adjusting the position of the telescope in two directions instead of just one (and at a slightly different speed in each direction!). This isn’t as difficult to do as it sounds, IF the mount moves smoothly. Many alt-azimuth mountings used on small refractors and short-tube reflectors come with two manual slow-motion controls to make the adjustments easier. Typically, to observe with an alt-azimuth mount you put the object at the edge of the field of view, observe it as it drifts across the field, then use the slow-motion controls or nudge the telescope to re-position the object before it drifts completely out of view.
The telescope that will give you the most bang for your buck is a Dobsonian. This is a Newtonian reflector mounted on a low-to-the-ground alt-azimuth mounting: the overall appearance resembles a cannon. Because of the way the telescope is balanced over the mount, the whole assembly is inherently stable and relatively free of vibration problems. You adjust the position of the telescope by nudging it gently; a good Dobsonian mount moves easily when you touch it so it’s easy to adjust the telescope’s position.
If you get a Dobsonian and later decide you’d really rather have a mounting that tracks, you can buy special platforms to put the scope onto convert it into a type of crude equatorial mounting. They’ll keep an object in the scope’s field of view for about an hour before the platform has to be re-set. But those platforms are expensive; a good one will cost more than your scope! But it’s an option you can consider adding later if you find yourself doing a lot of observing at high powers and are getting tired of the “nudge-look-nudge” routine.
One nice thing about amateur astronomy is that an initial investment in good equipment pays off, because telescopes don’t wear out. It’s not a hobby that requires you to keep upgrading; instead, you can use your funds over the years to improve the usability of the equipment you already have, or if you’re happy with the setup you’ve got you can observe for years with no additional outlay of cash.