It is essential that the tripod is sturdy and does not wobble, otherwise the object you are viewing will move about when you don't want it to, and also that the mount that goes on top of the tripod is of a good quality and is easy to use.
As with telescopes, there is a wide range of mounts available from cheap alt-azimuth mounts to expensive computerized GOTO mounts.
The type of mount you go for depends on what you intend to do and, perhaps more importantly, your budget.
It is important to not skimp too much here though.
Alt-azimuth Mount Alt-azimuth mounts are the simplest design, having two axes: altitude, which lets you move the telescope up and down, and azimuth, which lets you move the telescope from left to right (horizontally).
The trouble with alt-azimuth mounts is that in order to track objects in the sky, you need to move the telescope in both the alt and azimuth directions.
If you're doing this by hand it can be a bit tricky, especially at high magnifications.
Equatorial Mount The other type of mount available is the equatorial mount.
This type of mount also has two axes, but one of them - the polar axis - is tilted so that it lines up with the tilt of the Earth's axis.
This means that it can follow objects as they move across the sky much more easily than an alt-azimuth mount.
Motorized Mounts Motorized mounts enable your telescope to track objects automatically, and if you go for a GOTO mount, you will be able to go straight to an object and then track it.
All you need to do is to enter the name of the object into a handset, and the mount will point the telescope at the object.
The advantage of a GOTO mount is that you can find pretty much anything in the sky; the disadvantage is that you miss out on the enjoyment of trying to find a particular planet, star, nebula, galaxy, and so on, which is part of the fun.