What DVD/VCR combo would you recommend?

If you have a lot of tapes, maybe a VHS/DVD recorder would be worthwhile, to convert the tapes to DVD.
Samsung DVD-VR375 $179 at Best Buy

Sony RDR-VX560 1080p Tunerless DVD Recorder/VHS Combo Player $159 at Amazon

Regarding the multiple remotes – I’ve had good luck with universal remotes. The originals that came with the TV & stereo are in a drawer, I just use one remote.
I use the Sony RM-V310, you might want one up or down the line from there, depending on how many devices it needs to control.
http://www.sonystyle.com/webapp/wcs/stores/servlet/CategoryDisplay?catalogId=10551&storeId=10151&langId=-1&categoryId=28857&N=4294966064&accessory=true

I don’t like combination players. If one breaks, you have to replace both.

I concur with @filmfann. You can buy a decent DVD player for $40. You won’t be happy with a combination VHS/DVD recorder because they won’t dub commercial VHS tapes that are Macrovision encoded, which are the bulk of what people tend to have in their collections.

Update: the cleaner didn’t work. (Couldn’t get the player to recognize that there was a disk in it, so…)
Meanwhile, the kludge has grown to absorb my computer. We have our movies, but at a terrible price.