Psalms 1, 2, 22, 23, 25, 32, 40, 91, 100, 110, 130, and 145 are current favorites. I prefer singing ones that are directly quoted by the NT in reference to the person and work of Christ.
I am a musical car wreck so I go with simple tunes that I've written in the back of my TBS-
The Psalms of David in Metre (Scottish Psalter). You could start with
Crimond,
Dundee,
St Magnus,
St Columba,
Arlington,
Irish,
Winchester Old,
Azmom, and
Martyrdom. In a pinch you could use--gasp--
Amazing Grace.
You can find more tunes
here.