The chord progressions you want will depend on the sound you're after. You can do quite a lot of creating songwriting without ever straying from the basic "in-key" chords: I-ii-iii-IV-V-vi-[vii], and if the Top 40 charts are any indication, you can make a pantload of money with very "safe" chord progressions. However, if you don't want to be safe, and want to create new sounds,
Beyond this basic framework, there are some old standbys which work very well in particular places:
- Suspensions. These work very well as variations of the naturally-major I, IV and V chords of a key. In any of these, a suspension will want to resolve back to the major chord it's based on, and so these can provide needed movement within a "block" of this chord; suspend it, then resolve back to the major. A suspended I chord, with the fourth and octave, will naturally want to resolve to the IV chord which also has these notes, while the suspended V chord, with the fifth, octave and second, will strongly resolve to the I chord.
- Sus2 or Add9 chords. Similar to sus4 chords, sus2s work well as variations of I, IV and V, and resolve similarly to their upward-suspension counterparts. They have gradually become accepted as a minor source of "leading dissonance" in chord progressions; this dissonant major second becomes a more consonant note of the next chord, usually either the root or third. The IV chord works very well as a sus2/add9 variation, because the second of this chord is the fifth of the native key, and so can resolve very naturally either to the V or the I chord. Guitar players will commonly use a Cadd9 chord fingering instead of the open-position C because the transition is very easily done from the root G chord.
- Dominant 7th chords. These work very well as a variation on the V chord, because the dominant seventh of that chord is the perfect fourth of the overall key, so the entire chord will naturally resolve to the tonic. For instance, A7 in the key of D has the notes A-C#-E-G, and with the seventh, second, fourth and fifth of the key in one chord, every note wants to resolve toward the tonic chord. Dominant sevenths also work well in the vi position; in the key of G, an Am7 has the notes A-C-E-G, and so will resolve handily to the IV chord of C-E-G. It's less often seen as a replacement for the IV chord, because the dominant seventh of the chord ends up being a minor third from the tonic (C7 = C-E-G-A#), but for that reason, if you want to transition from major key to minor key without changing your root note, V-IV7-i is a good way to do it.
- Major II and III chords. The major II chord is theory-based as the "V of V". In the key of G, D is the V chord, then in the key of D, A is the V chord. Progressing I-IV-V-II, then using the II as the new tonic chord from then on, is an easy way to change keys. The major III chord is a definite rule-breaker because it augments the fifth of the native key, and you generally try to avoid modifying the root, fourth and fifth of a key in your progression. However, III-IV-V (for instance F-G-A) forms a powerful progression leading back to the I chord (D).
Beyond these basic tricks, look at the chords as a movement of voices. Figure out where you want to start and end up, write the notes on a staff (it does help to be able to read music, even for instruments like guitar and bass that rarely use it), then see where each note of that chord can move to transition from one to the other. Most times, your "transition" chord in the middle will only involve one voice moving half a step in either direction, then the chord will resolve as one or both other voices in the triad make a similar move. Don't forget your inversions; if the movements seem too large, you're probably trying to move to the wrong inversion of the chord; drop the top note of one chord by an octave until the movements work better.