Since there is fret hand movement on the D or 4th string you'll want to fret so that the fingers playing the notes on the 2nd and 3rd strings don't need to reposition as the chord shape changes slightly. That means you'll want to play the initial A minor chord with the 1st finger on the 2nd string, middle finger on the 3rd string and 3rd finger on the 4th string. Then when the chord moves up to the 3rd and 4th fret on the 4th string you can simply slide up your 3rd finger or add your 4th finger without having to shift the others.
Your picking had should be assigned to the top 4 strings on the guitar. Thumb takes notes on the 4th String and so on up to the 1st string which your ring finger will be assigned. Each picking hand finger has its own string. Don't sweep strings with a single finger! Play all notes as straight 8th's and pluck as smoothly as possible while letting the chord ring throughout the measure.