The quickest way to start increasing the number of chords/inversions you have at your disposal is a simple two-step process. First, take any chord you already know (start out with simple chords, like a major triad), and have a look at each of the tones played when the chord is struck. For example, your standard first position C chord is composed of the following:

---
-1- C (R)
-0- G (5)
-2- E (3)
-3- C (R
---
C Major
Secondly, and lastly, slide each note up to the next highest tone in the chord (alternately, you can move them all down the to the next lowest tone -- but not if the chord is in first position as in our example). Now, aks yourself: can you string all those tones together into a chord and play it? If so, you've got yourself a new inversion for your chord! Do this as many times as are useful per chord. Sliding our C Major chord up the neck gives us the following inversions:

-----------------
-5---8----13--
-5---9----12--
-5---10---14--
-7---10---15--
---------------

C/E*  C/G   C
Repeat this for every chord you know, practice playing chord progressions with the new chords for an hour every day, then after a couple months go look up more complex chords and do the same thing with them.

* This notation refers to a C chord with an E in the bass. This can be used for inversions (as in this case), or for non-chord tones, such as in A/F#