To transpose chord sequences: move every chord in the song up (or down) the same number of half steps. Change only the letter name (e.g. Eb to C), retain the type of chord (e.g. major, minor, 7th). The basic knowledge needed to do this is knowing the chromatic scale and knowing how to count -- by which I mean you don't have to know about mediant triads or Doric modes to simply transpose chords.

To transpose chords often and more easily, use the number system previously mentioned. For example, the typical "three chord" song in the key of C will have the following chords: C, F, and G (or G7). These are called the I, IV, and V (or V7) chords in the key of C because c, f, & g are notes # 1, 4, & 5 in a C major scale. To transpose to the key of X, use the I, IV, & V chords in the key of X, which are named after notes # 1, 4, & 5 in the key of X. For example, if X is the key of G, the chords would be G, C, and D, because the 1st, 4th, and 5th notes in a G major scale are g, c, and d. To use this system, it really helps to know the major scale in the keys used, and/or the formula for constructing a major scale (it's fairly simple if you know the chromatic scale, the formula, and can count).

Next I would suggest finding a booklet that lists common chord sequences. I see them at almost all music stores. Getting familiar with typical sequences, such as I-IV-V and I-IV-VIm-V, can help a lot with both working out the chords to a song you hear and getting practice transposing.

It's great to be able to play in a number of different keys, but unless you want to master every chord in every key -- a HUGE task -- you'll find the capo helpful. Playing in the key of Db is a whole lot easier when playing "key of C" forms capoed up a half step than when playing "key of Db" forms.