Monday - Get a trail map, study it good, don't venture too far from civilization. Some trails can connect with others that lead even further from the pump, and before you know it you could be in bfe with no friends and no fuel. YNP should be self explanatory since they still guide you through it IIRC.
Riding on Friday - A lot of folks talk about Jefferson area. IMHO, especially if you have 'rookie riders', Jefferson is DEFINITELY NOT the area even a skilled rider should be right now unless you have the self discipline to stay off any of the bigger stuff. It is also still extremely rocky back there, so the chance of carnage is quite high. I would stick to some of the lower trail systems and try to find any powder stashes that may be left.
You should be able to find something for everyone to have a good time. If the snow was better, your options would expand 10-fold for that area. Good luck and ride safe.