Check rubicon.com
Ideally, you would be best off hooking up with the TLCA Rubithon the 3rd week of June. It'll be the 20th Anniversary, and it's the first group run of the season.
Time wise -- Depends on the rain -- locals usually shoot for Memorial Day, but the last few years have been too wet for that early. Latest is Sept/Oct -- After that you're snow wheeling. Late Sept gets cold at night.
There are locals that snow wheel, but I'm not sure that would be my first choice for a rookie on the 'Con.
While people can make it through in a day, you really have to plan spending the night on the trail -- Usually, people drive from Loon to Buck, camp over night, and either go to the springs and Camp, or continue on out. It's a long set of days doing it in two days. The thing is, once you're there, you want to stop and soak in the surroundings.
If you decide to go solo, you want at least one other rig that has done the trail You can get lost. There are a few spots that are tough to negotiate, and even more difficult at night.
If you go later in the summer, you also need to keep in mind that Jeep Jamboree (Jeeps only) and Jeepers jamboree (open to all) are issued parade permits, which effectively closes the trail for two weekends. If you do Jeeper Jamboree, I think the cost is ~$260. TLCA's Rubithon is $100.
A good source of info is also on pirate4x4 -- go to the forums, and then there is a rubicon trail page.
Last thing -- Are you going to bring the TRD FJC, or do you have another rig? If you bring the FJC, prepare yourself for body damage, a lot of spotting, and high stress levels. It will absolutely take you 3 days to run the whole trail in the FJC.