Don't let any die-hard bluegrass mandolinists see this thread. Open chords are sacrilege to them let alone letting them resonate.

I started with mandolin and I've got some good chord cheat sheets. What is beneficial to know for beginning fiddlers is that there are a handful of easy "one finger" and "two finger" open chords. These are easier to play without the dreaded accidental string muting screeches.

Coming from mandolin, it's my experience that people often misunderestimate how much is applicable between the two instruments. Same exact tuning, you finger out the same notes with the left hand(albeit with frets). Though my mandolin is oft neglected since I've taken on the fiddle, I'll often try new melodies, rhythms, and passages on the mando before picking up the fiddle and how.

Tremolo is the mandolin counterpart to the violins vibrato but even more important if you want to get that simulated sustain, especially on classical pieces.

My recommendation is to buy more than one, all in varying thicknesses and flexibility. You may find some are easier for tremolo than others. I tend to prefer them on the thinner side.

Gordon Shumway said

That reminds me that I've got a lot of guitar picks in different thicknesses, but I've never used them, and I don't know if mandolin picks are different. For the purposes of good tone production, I found that the thicker the pick, the better, so I don't know if I kept the thin ones.  

A lot of mandolinists prefer thicker but I'm the opposite. I think I dig in more and strike harder to mask any significant projection differences.

The picks are the same. It's all really personal preference. I've found some picks are easier if a lot of tremolo is involved and some are better for high tempo melodies.