When I have either of the following connections, my TV picture is clear:
(cable signal direct from wall, no cable box, using regular coax cable)
Cable --> TV
Cable --> DVD player/recorder --> TV
Cable --> Splitter --> TV

But when I do the following set-up, my TV picture has noise:
Cable --> Splitter --> DVD player/recorder --> TV

The reason that I set up this connection is that I will be adding a 2nd DVD recorder to the setup to record conflicting scheduled shows.

I have tested all the cables individually using the simplest setup (Cable-to-TV) and they all work fine.

Is my new setup too complicated for the signal to pass thru properly? Will switching to RG6 cables solve this? Any ideas? Thanks.

You should get a powered splitter.

Any suggestions where to get one? I looked at monoprice but couldn't see any (the only ones I saw were for PC Monitors).

Try radio shack. Maybe firefold? Our cable company gave us one, so I'm not sure where he got it. eBay probably has some too, and I may have seen one at Best Buy once.

A few popped up just by googling "powered tv cable splitter."

Try "coax distribution amp" instead, it'll be easier. And look for Channel Master.