What is a good way to handle a multi-plane adventure?

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.

I'd suggest changing some of the following to get a unique flavour for every plane:

1. Temperature - a basic change, but if you require players to constantly find water it has a big impact on the game. If they need to melt ice a lot for something to drink, they will hit a problem when the fire attracts monsters. Describe their breath frosting the air and the chill in their bones, or the sweat dripping down their faces and how their metal armour burns them. In extreme situations, they'll need a form of protection from the searing heat/chilling cold. These sorts of problems can be a small annoyance if they happen once, but a whole plane of it will make players feel very far from home.
2. Colours - a purple sky, green fog everywhere, orange soil. A vivid description of a strangely-coloured plane can work wonders. Players again notice that they are in a different place. Don't use this too much if you tend to use it to show magic places on the Prime Material Plane, and don't overuse a single colour ('The water, the sky, the dirt, even your waste is bright blue!') unless you have a very good reason, or the world looks one-dimensional.
3. Ecosystem - on the Prime Material Plane, energy comes from the Sun and is absorbed by plants, which fuel the animals. Now imagine if instead, animals can gather energy directly from the air/earth/native energy... this would mean no herbivores or plants, and cause a large shift in the environment. There wouldn't be any plant cover for ambushes or concealment, and camouflage would be changed too. Also, if you need some monsters, just get the stats of a standard creature and tweak them a little, then give it a custom appearance and role in the ecosystem of the area.
4. Laws of Physics - this has been covered by the other answers, but bears repeating. Laws of physics are immutable constants, so changing them really has a big impact on how a world feels.
5. Magic - I urge caution when using this, but changing spells - even in small ways - can cause spellcasters to freak out. Few players will forget the world where good old Magus' spells suddenly halved their duration and doubled their strength, or where Zantar accidentally summoned a badger while trying to get an air elemental. The key is to not nerf the magic-users too much or to only do it on one plane. If you want to mess with them, have high-magic and low-magic planes where spells work at various levels of effectiveness. If you want a more balanced setting, try to have minor changes (e.g. switch water and fire spell effects around) or slightly beneficial/harmful effects (e.g. healing is accelerated/spellcasters take twice as long to memorise spells). Don't forget to have a huge variety of ways the locals utilise magic.
6. Senses - major changes like all natives seeing only in infrared make a plane stand out. However, this only works if you use it sparingly - if every plane is weird beyond belief, players stop recognising the differences ('Ho-hum, another world in which the natives use a different light spectrum.'). Other possibilities are worlds in which sounds don't work properly and players can either sneak better or detect others sneaking. Don't forget that any modifiers for the players should apply to the natives as well, unless they have specifically adapted around it. If possible, make the changes have at least a small impact on the mechanics - tactically-minded players can often use anything as an advantage, so why not humour them once in a while?
7. Intelligent Groups - this will usually be none, one, two or many. No intelligent groups means that there won't be anything beyond monsters and animals on a plane. This can be boring, or make a nice change from fighting evil empires, depending on the situation. If there is one group, it will be like Earth (with only humans). The race will be made up of several factions, unless there are so few members (or the environment is so dangerous) that they all need to work together to survive. Two groups will either fight or be allies, in general. A good sandbox setup is a powerful ruling race and a more numerous but worse-armed and militarily weaker servant race. The PCs may decide to help the better-paying, more powerful race or the weaker race. An interesting twist to this is making the servant race be evil (like drow or duegar), which the stronger race has bound to prevent more wars and death. If there are many races, a situation like a standard D&D world tends to develop with good and bad races, and some in the middle. Alternatively, a world full of good guys makes a good 'home plane' to access other planes from, while a world that is evil to the core can make an awesome sandbox - the PCs have valid reasons to butcher their way across a whole continent, which in my DMing experience maketh the player's heart glad.
8. Prime Material Presence - this covers how many of the species and intelligent races are found on the Prime Material Plane. A plane with only elves, for example, can be very interesting to create and explore. A world like the Prime Material Plane but missing all mammals can be more interesting than a world made from scratch. Also consider having, for example, some powerful wizard leave their home to set up a quiet place for study and teaching in a corner of some uninhabited wasteland. Now fast-forward 500 years for an insta-dungeon! Alternatively, there could be a clan of some sort who found a portal and entered it to find the plane the players are on. 200 years later, all that is left of their origin is their language and some old myths. Very interesting for players to discover and maybe help.

I will add more as I remember of it

I think one of the easiest ways to play on alternate planes of existence, but also the most boring way, is to just have them be "other countries". It's a common solution, especially for high level characters, when the familiar cities and environments they grew up in aren't challenging any more, and you want to introduce newer settings where todos is higher level, to match the characters.

The problem with that is that it misses out on the potential for otherworldliness. An extra-planar city shouldn't be just another city - albeit a weird, outlandish one where the people are blue-skinned or the currency is crystal shards. It should be really estrangeiro.

How do you do that? Try to mess with the basic assumptions of the world to make the players feel the weirdness. Make physics behave differently, for instance, making them constantly think differently about their actions.

A nice idea to borrow comes from the novel Duna (yes, yes, of all places), where a society with force-field technology eschews brute-force hack-and-slash tactics, and duels are resolved by feints and misdirections allowing a slow, slow knife in through the forcefield. Try that in your game - change the combat rules a little bit so that any slashing or bludgeonning weapons, or weapons doing 1d8 or higher (for instance) have their damage massively reduced, forcing players to think differently. Make sure to describe it in details.

Take more ideas from novels about very alien places. Another favorite of mine is Roger Zelazny's Âmbar novels. Take the Courts of Chaos, with its ever-shifting skies and wibbly-wobbly geography, and how no-one can andar anywhere, but you need magic to teleport or summon bridges. Have the players be helpless at first. Make them feel like they're somewhere totally different.