Both cheap and mild are in the eye of the beholder.

Two possibilities:
Start in your local store, try Merlot or Shiraz wines in your budget.
While both Merlot and Syrah (Shiraz) can make big powerful reds, on the
low end they tend to light.

Try a hybrid-grape blend such as Taylor Lake Country red. They tend to
be light and a little sweet, and I believe lambrusca and hybrid grapes
have higher levels of resveratrol. If still too much for you, try
diluting with water or seltzer.

If one of these work for you, in a few months see if regular
consumption has made you more receptive to wine, and maybe then expand
to more "serious" wine.

I could give you lots of recommendations for lighter wines (Beaujolais,
Loire, CA) in the $10-15 range, but think if you're drinking for health
reasons alone you probably will be happier with a under $6 per 750
wine.
wrote:
> Hi
> I want to start drinking wine with dinner for the health benefits, but
> I'm not a real fan of the taste. Can someone suggest something thats
> cheap and mild? Thanks.