Thread: At / this / next / last weekend

На эти выходные поеду к моему дедушке = for the whole weekend (Saturday + Sunday).

В эти выходные поеду к моему дедушке. This would mean that you're planning to go to your grandfather on one of your day-offs, usually Saturday or Sunday. I'd personally say it a bit different though. На (этих) выходных поеду к дедушке, but I think it's all alright.

Originally Posted by iCake

I'd personally say it a bit different though. На (этих) выходных поеду к дедушке...

Oh, I forgot this case.

George1992

Note the difference between "на выходные" and "на выходных".
"на выходные" = "for the whole weekend". It is "for the period of time".
"на выходных" = "in some (maybe not known yet) moment of time during weekend". This is the same as "в" in the examples above!
Grammatical case rules the role of the word in the sentence here.