Of course it will first be necessary to completely memorize the dorian mode pattern. Practice the mode slowly and accurately, both across the neck, and up a single string. Be sure to play the mode forwards and backwards.
It's important to start blurring the lines between the major scale shape and the dorian shape on your fretboard. Since the major scale and dorian mode starting on the second degree of the major scale have all the same notes, you should try and start viewing them as one scale. To start getting comfortable moving back and forth between the major scale and dorian positions, practice the pattern outlined above.
The idea is - you play the ascending G major scale, then move up to the A dorian position (same notes as G major), and descend in that position. You complete the scale by returning to your original position to play the final note "G". After you've mastered this, you can take this concept to another level. Try starting in the major scale position, and switching up to the dorian position on one of the middle strings, all the while maintaining your tempo and flow. You can try something similar while descending.
Once you've got the scale under your fingers, you can start trying to improvise using the dorian/major scale patterns. Download this 1.5 minute mp3 clip of me strumming an A minor chord and practice playing your G major/A dorian scale positions over it. Then, try making up licks similar to the ones presented here by Santana and others. Spend a lot of time with this - be creative. Try mixing A minor pentatonic, A blues scale, A dorian, and any other minor scales you know into your solos - don't feel like you have to only play one scale throughout!
By the way, don't worry if your solos don't sound great at first. Getting comfortable with a new scale takes time, and certainly won't yield wonderful results at first. That is why we practice - so by the time you're playing it in front of others, you sound top notch!
If this whole modes concept is fuzzy to you, don't worry too much about it. Just practice, practice, practice, and chances are, you'll stumble upon the logic of modes yourself. Try not to get frustrated if things aren't "clicking" - they will with time.
