This was the subject of a recent discussion on the Woodturner's Resource site.
In order to use collets, you must have a collet chuck. There are two 'budget' options for collet chucks - the PSI option and the CSUSA 'Apprentice' option. Both chucks screw directly onto the lathe spindle - the PSI option is dimensioned for a 1"x8PTI spindle, and comes with a 3/4x16TPI adaptor, while the CSUSA chuck is dimensioned for a 1.25"x8TPI spindle and comes with an adapter for 1"x8tpi. Both are made in China.
There are other suppliers of screw-on collet chucks - notably Beall. The Beall chuck is smooth and must be tightened with a spanner wrench. The PSI and CSUSA chucks are knurled, and it is often possible to tighten them sufficiently by hand; both come with 'tommy bars' if you really need to squeeze down on a piece. There are also chucks that mount using a morse taper.
The second consideration is to decide between imperial or metric dimensioned collets. A critical point to understand about collets is that the nominal dimension of a collet is the maximum diameter it can grip, and most collets (both metric and imperial) have a working range of about 1mm. The PSI and CSUSA chucks come in sets that include five imperial collets in 1/8" steps between 1/4" and 3/4". For many people that's good enough - you just turn whatever you are making to have a matching diameter. But you can purchase imperial collets in 1/64" increments if you need to fill in the gaps, or you can purchase collets in metric dimensions in 1mm increments.
I have the PSI set and have been very happy with it. I have not found having gaps in the clamping range to be a problem - YMMV. I separately purchased two additional collets - 1/8" and 20mm so that I have a slightly broader range of options. Incidentally, the two additional collets came from China and seem to be fine. I'm sure that US made collets would be better (and more expensive), but I'm not a professional machinist and I can't justify the cost of something more expensive for my occasional use.
There are two helpful documents in the IAP library that discuss collets and collet chucks -