This is a question asked by somerled in a topic that got locked in the Flat Earth Theory forum.
I thought it was a good question and should be addressed and this appears to be the right place to address it.
The easiest planet to see rotating is Jupiter. Jupiter has a day of less than 10 hours, so on a long winter's night you can easily witness it do a full rotation. You will need a decent telescope. Let's make no mistake, your not going to be able to detect it rotating with binoculars or even a modestly priced 4". However, with a well collimated and quality 6" scope or bigger it should be no issue at all to easily see enough detail (like the GRS) to see it rotate.
Mars is probably the next easiest to see rotating. It's day is just a tad longer than earth, so you won't see it do a full rotation in one evening of viewing, but you could certainly see it rotate through 180 degrees or more. And if you observe it every night over about 3 weeks, you can get to see the entire surface. Again, a decent scope will be needed for this and if you're a skilled observer you could do it with an 8".
The next (and probably last) would be Saturn. For this one you'll need at least 10" scope to see enough detail to detect rotation. It's day is just under 11 hours, so it should be easy enough to see the entire surface in one (winter) night.
Although Venus can be fairly close to earth from time to time and easy to see, it appears featureless and detecting it's rotation is beyond what most amateurs can muster. If you have a big scope (10"+) and a good set of filters, you might be able to see the notion of a bright spot, but that would be seeing the motion of the atmosphere, not the planet. A day on Venus is about 5,800 hours. So, even if we could see the surface features, it might be tough for the casual viewer to detect rotation. The atmosphere, however, circles the planet in under 100 hours.
Mercury is tiny and far away. It's day is also very long - nearly 1,408 hours. I have access to an 18" scope and have never been able to even see details on Mercury, so I'm going to say it's not in the realm of reasonably easy to record rotation on Mercury for an amateur.
Uranus and Neptune are quite far off and even quite large high quality scopes have difficulty detecting any features, so again not in the realm of the casual viewer.
That being said, there are earth based instruments that can detect the rotation of all the planets. It's just that you or me are likely never going to get to play with them.