The key is knowing the movable shapes, that is, chord shapes where all 4 strings are fretted. A simple shortcut is to use the shapes you know, but reserve your index finger to barre the strings at some fret.
Example 1: Make an F chord, but use your middle and ring fingers to press down the frets. Now, slide them up one fret and use the index finger to barre the first fret. You've got an F#. Slide the whole thing up one more fret and you've got a G. Two more frets (so your index finger is pressing down at the 4th fret) and you've got an A, and so on.
Example 2: Make a C chord, then slide up 2 and barre across the 2nd fret. That's a D. Slide up to the 4th fret, and it's an E chord.
Example 3: Make an A7, slide up 2 and barre, and it's a B7. Slide up to barre the 5th fret, and it's a D7.
I'm sure you can work out more from that. It helps to know which string holds the root note of the chord.
Something else to look at when you're doing this is the relationship the I-IV-V chords have near each other. For example, in the key of D, you can use the A/Bb shape barred at the 5th fret for D, the F shape barred at the 4th fret for A, and the C shape barred at the 7th fret for G.
I hope this helps. Play around, and have fun with it.