The heaviest string, the 6th, is the root note of the chord so you can thump it and then strum the strings to get a nice sound going. But playing G,C,G,D starts sounding very repetitive because the sound changes registers (gets significantly higher or lower) for the different chords. You play G open, then C on the 5th fret so every single note of the C chord is 5 frets higher than the previous G chord. This is unlike the guitar or banjo where, if you play a G down-the-neck and then a C down-the-neck, some of the notes are the same and all the others are close to the previous chord's so it all sounds similar.
In order to achieve this similarity of sounds on the dobro, you must change the strings you strum. If you strum/pluck the bottom three (1,2,3) strings for the open G, then you should be pluck/strumming the inside ones (2,3,4) when you move up to C and D (5th and 7th frets respectively), and if you want to end on the G chord at the 12th fret, you want to be plucking even farther in, using the 3,4,5 strings.
To get a minor chord, some folks will retune, dropping the 2nd string down a fret/half-step to Bb. Then they play the minor chords on the 1,2,3 strings and the major chords on the 4,5,6 strings (changing bar position appropriately to keep the chords in the same register). Another way is to NOT play the minor string (the 2nd and 5th) at all for the minor chords, basically turning the chord into a Power chord that doesn't distinguish between minor and major.
Additional partial chords can be made by angling the bar and just plucking certain strings, but once you get beyond the major chords, the dobro is not so useful for chording and really needs to be approached as a melody instrument (like the fiddle or saxophone).But for basic I, IV, V songs, the dobro has a full guitar sound without the player having to learn much more than a fret number.