Depeche Mode and R.E.M

R.E.M. get "Reckoning", "Murmur" and "Out Of Time".

You have a good start with your REM collection. For further REM selections, I would go with:
1) Murmur
2) Reckoning
3) Document
and then,
4) Fables of the Reconstruction.
They have released a few of them on Mobile Fidelity.
Sorry, can't really help you with the Depeche Mode, but I am sure someone here will be of help.
Enjoy the music!

Don't forget DM's violator!

I have been a REM fan from the beginning and the previous comments are good suggestions. I remember seeing them open for the Swimming Pool Q's in 1980 and thinking that they were going to be big someday despite the lead singer dancing like he was having an epileptic fit. The first 4 or 5 albums are the best. The albums after Bill Berry left are not as focused. I think that an often overlooked album is "New Adventures in Hi-Fi". It is an electric album that was recorded live during soundchecks. The band is really tight and Stipe's voice is very strong. One of my favorite REM songs is "Binky the Doormat." Peter Buck has been playing with Steve Wynn from the Dream Syndicate in The Baseball Project which is also a fun band. Buck has a solo album scheduled for release this Fall.

If by chance you get the CDs of DM, make sure they are the remasterred ones from Rhino. They sound really good.