"When working with a mixer and a multitrack recording system, how much quality difference is there between routing signal through the busses to the multitrack versus using the direct outs? I like to use the busses on my mixer because using the direct outs requires a lot of patching."

It depends on the mixer. As you seem to know, but for the benefit of other readers, routing signals through the subgroups versus using the direct outs has them go through several additional gain stages where you run the risk of adding unwanted noise and distortion. The old analog mixers that everyone holds in such high esteem are often not that clean and quiet. They sound great, but... On those you are definitely better off to do what it takes to use the direct outputs. The best way to handle this is to wire them through a patch bay system so you can easily reroute the signals as needed. If your mixer is a more modern analog design, you can probably get great results either way. If your mixer is digital, it shouldn't matter at all. You should also consider a few high quality outboard preamps in the future. Running a mic directly to a preamp and then directly to a recorder often gives you the best overall quality since the mic preamps in most mixers are not ideal for all situations. Just one or two outboard preamps can make a big difference because you can use them each time you do an overdub.