Wudijo also made a nice video about DH in groups (link), which is for SC but it might still be a viable option (though not the best). Last but not least, a class/build that is really underestimated at the moment is the Hammerdin crusader. You can see in 2 player that it's slowly picking up and showing its potential.
Basically, you and your friends can pick *any* class. Also keep in mind that switching classes takes close to no time in Diablo 3. So here's the breakdown:
Barb: Is mandatory support in 3 player and 4 player speed GRs after 80+
Crusader: Is an option for speed farming or in lower groups, probably a good DPS choice for the barb which will feel useless in GR50-GR75 gem upgrade rifts
DH: Is the #1 for Torment speed farming and everything up to GR50, and with the new Grenades build an option for lower speed GRs as well.
Monk: Mandatory support monk in any group (2p, 3p, 4p) after GR80+ (maybe even earlier on HC). A second, generator single target DPS for super high pushing at the moment. But you can probably get away with not playing it until the last week of the season.
WD: Firebats Arachyr WD is currently the #1 area damage DPS class in high GR pushing, but for speed wizard is probably a bit better (more mobility).
Wizard: Has several good speed farming builds (Archon with Firebird, Archon with Tal Vyr Manald Heal, Archon LoN for T13 and low gem ups) and even a support build.
Choose wisely! (Hint: "wisely" here means "the class you like".)