The dream sat station of the ham is composed by a circularly-polarized yagi array, some dishes for the upper bands, very low noise mast-mount preamps and a computer controlled AZ-EL rotator.
In the case you are interested to build (i.e. BUY) this station type, start selling your car....
If instead you are penniles like me, you don't have to renounce to sats, just to accept a little lack in performances...
A good starting point for sat operations are LEO (Low Earth Orbit) satellites. They are orbiting at 200/300 Km from earth surface and there is a passage every 90 minutes more or less. You just have to pick a sat tracking program for your PC and updated Keplerian Elements (Look at Amsat-NA). Then, take a dual band HT (who do not have one?), program the frequencyes and go on!
For an easy start I suggest you to try UO14. There is an helpful guide to the so called 'Easy sats' that you can read at ########, where are described modes, equipments and all the things you need to become an expert operator. I don't promise that you will do a QSO at your first attempt (the operation stile is contest like...), but with a little effort you will do it.
At this point some of you will say: "And antennas? Do we have to use the HT's whip??" .
Antennas are -I think- the most important part in your sat station.
In this kind of operations is very important (if you have a fixed antenna), the form of the radiation lobe.
I counsel you to start with a normal dual band vertical (maybe you have it!): it has a very bad radiation lobe (Fig.1), but it's a quite inexpensive initial antenna; it also will show you the real possibilities of easy sats. I have made my first sat QSO on UO14 using this setup!
For a little more performances you can use omnidirectional antennas: a good starting point for experimentatoin are the so called "Egg beaters" antennas and all their variations. You can find some projects in my links page going to K5OE's site.
For a much more flexible setup you can use an "Arrow", or something like, antenna: weightles, portable and with high gain and directivity, it is made by two yagis scrossed on the same boom; you just have to follow the sat in the sky with your hands, so if you don't have a vox system in your HT, it wuould be useful a little tape recorder: you will compile your log after the pass, hearing the registration.
Good luck and good QSOs on LEO sats!