Cutter, before I start rattling an answer 'round my bone-box, I have to suggest you look at the 'Planescape' setting. It was released for Second Edition AD&D, it's not hard to find on eBay, and it's the best examination of planar adventuring a basher could ask for. Even if you don't want to make use of the setting and cosmology it describes, the sections in the Planescape boxed set on how to GM a Planescape game offer plenty of advice that might be useful in other planeswalking campaigns.
Now for my answer. For a GM, the most important bit of advice I can give is to make another plane be another plane: Not just another place, but another reality, one where the natural laws that the players don't usually think about work differently. You can be obvious about this (changes to gravity are a classic) but more subtle changes can be particularly fascinating and alien. (Let players know that their reflections are lagging a couple of seconds behind, and they'll stare into that lake all day.)
Players, look for an learn to identify the differences between the laws on your home plane and the ones you adventure on. More importantly, learn to exploit them: Subjective gravity is a canny tactician's dream.
The second piece of advice I can give to a GM is, well... You know all those weird and wonderful monsters that your wizard's been summoning? The ones which are clearly out-of-place on the Prime Material due to obeying different natural laws, made of elemental matter and/or being tied to the nature of the very alignments themselves? In an extraplanar adventure, those critters will be right at home. They're not just weird supernatural creatures called into the world by powerful magic, they're part of the local ecosystem, part of local society, and (if they've the bent for it) part of the local politics - and a fiend with friends is the most dangerous kind. IN the planes, context is everything.
Players, in an extraplanar adventure, most of the critters you encounter will be on their home turf. On the prime, they're out of their element (sometimes literally) - but on their home planes, they've got a niche, they know how things work, and - and this is important - they have the home team advantage. Planar natives that you encounter will know and understand their environs and know how to use them against noisy intruders - and might be willing to offer their services as local guides and experts for the right price.
I'll add more to this answer when I get the chance.