Particularly Major 7th chords. I've been messing around in the key of E major and have decided to start of the bass line of this bit of experimenting with the Emaj7 chord and I'm trying to resolve to some other (more mellow) chord. After messing around a bit, I've found a few nice ways of resolving it (Em alone seems to work alright), but I'm not really sure if I'm going about looking for the chord's resolution properly. Is there even a proper way to resolve 7th chords?
Some chord sequences to explore starting with Emaj7, these may help you to start to develop a vocabulary of sequences that work:
Emaj7 - E6 - F#m7 - F#m+6 Emaj7 - F#m7 - G#m7 - F#m7 Emaj7 - Fdim - F#m7 - Gdim - E/G# - G#/F# - C#m/E Emaj7 - C#m7 - Amaj7 - E/G# - A/C# - B/D# - C#/F - Fdim - F#m7 Emaj7 - Amaj7 - B - C#m - A - B/F# - E
Obviously this is not an exhaustive list, and there are many more creative sequences available. Different periods of music had different rules (formal or informal) that were followed in the chord progressions but unless you are trying to compose in a particular style there is no best practice as such, though arguably some progressions will simply sound unnatural, unusual, awkward etc and will therefore be more suited to styles such as eary/suspense music than fanfares or pop music.
Chordal 7ths like to either resolve downward to the next tone in the key, downward chromatically, or be held as a common tone. If you want it to resolve downward to the next scale tone, to C# in this case, you could make the chords C#m(7), B9, A(M7), or F#m(7). If you wanted to keep the seventh as a common tone, D# in this case, you could use B(7), D#dim, or G#m(7).
There are more out there especially when you look at alterations, borrowed chords, and extensions, but this should you an idea where to start and what to look for.
To add to Dom's answer in the case of Dominant sevenths you are also going to need to resolve the leading tone. So for instance in E Major the Dominant seventh's notes are B, D#, F#, and A. The seventh has to resolve down to the G# and the D# (Leading tone) has to resolve to a E.
There is only two chords in E major with those notes. The tonic chord and the Sub Mediant chord.
Chords that include the tritone - in G7 that's the F and B - have a pull towards resolving the F to E and the B to C. Hence G7 as the "dominant 7th" of C major.
Where there is no tritone, though there's always an overall pull towards "home", the tonic note/chord, it's less insistent. You needn't think of a maj7 chord "resolving", more of where ot might move to.
If you're in E major, any E chord IS "home" harmonically speaking. If you find Emaj7 too astringent, you could move to E6. Or just to E, though the change in density may feel rather empty. "Planing" can also be effective, either diatonic - Emaj7, F#m7, G#m7... or chromatic - Emaj7, Dmaj7, Fmaj 7, Emaj7.