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.