Set aside uninterrupted playtime
The great thing about soloing is that you never have to worry about interruptions bothering anyone else but you. And if you have close friends or understanding guildies, you can usually impose on them with non-emergency AFKs as needed. But if you're going to PUG it up using Dungeon Finder, you should make arrangements to only be interrupted if absolutely necessary. Captain Obvious says it's inconsiderate to do otherwise. If you're a parent, I find that after bedtime is great.
Primetime is Dungeontime
If you wait until your server is quiet to queue up, you'll probably find your options limited and the queues long. So I highly recommend recording your favorite shows so that you can dungeon crawl when everyone else is.
Plan to spend around an hour
In general, old world instances are much longer than the 5 mans we run in Northrend. Of course, there are exceptions such as Scarlet Monastery Graveyard, which is an easy 15 minute run. Also, you may end up in a dungeon in which your whole group is unfamiliar and/or can't remember. And there's a lot to be said for enjoying the scenery instead of just rushing through.
Be a healer or tank
If you really don't like playing either role, don't take this advice. Playing WoW should be fun. However, if you do enjoy being a meatshield or playing whack-a-mole, then you will find your waiting time much more reasonable than queueing as DPS -- possibly even as instantaneous as it is at 80. Of course, some people queue as tank and then expect someone else to actually do the work. That's just stupid, since runs only go smoothly if everyone performs their role correctly. But I'm sure you all know that. Also, make sure you understand the basics of your class in your chosen role and have the appropriate gear for it at that level.
Get the quests
I have solo or duo leveled most of my characters previous to Patch 3.3 and therefore am used to just dropping the quests requiring instances. But now I seek them out. The quest rewards for dungeon quests are often better than the drops inside and are usually better than nondungeon quest rewards. And, of course, there is the nice chunk of XP you get for turning in quests. Picking up quests isn't always possible for every dungeon, particularly for the ones that are located in enemy cities. And some are hard to find. The really nice PUGgers share their quests at the beginning of dungeon runs... if they have a chance before the tank starts pulling.
Prepare your bags
Visit your bank, shop at vendors, mail to your bank alt and otherwise empty your bags of everything you don't need for the run. Then make sure you stock up on food, drink, bandages, potions, reagents and anything else you will need for a run or two. Make sure to repair and restock inbetween queuing. You can only trade conjured items across realms, so your groupies won't be able to help you if you're missing required items and you don't have a mage.
You can start using the Dungeon Finder tool at level 15. I don't know if I'd recommend starting that early -- level 18 might make the dungeons go a bit easier. When you sign up for a random dungeon, you will be put in a group of similarly leveled people in an appropriate instance.
Stay till the end
Defeating the main boss for the dungeon will give you achievements, great gear drops and most importantly, an XP bonus and The Satchel of Helpful Goods. Unlike the heroic dailies at max level where the best goodies only drop once a day, the satchel will drop every time you complete a random dungeon while leveling. The satchel contains sweet, delicious gear that is usually better than anything else you can get at your level. Even at lower levels, you get a pretty blue item to drape on your bod.
Don't stay till the end
The satchel and valuable playtime is not worth it if you're stuck with jerks and/or idiots. The players blaming everyone else for wipes are usually the ones who are causing them. And the ones who are ridiculously rude from the outset will not improve during the run. You'll enjoy your play session much more if you are questing while working off the Dungeon Debuff than if you are putting up with That Guy. I once stuck with a run where the "leader" made a rape comment in the first few seconds of my zoning in. Dumb move. I should have put him on ignore and scarpered. The funsuckers only win if you let them, so degroup from the jerks and don't look back.
Need everything you want
I know not everyone agrees with me on this, but it really reduces your stress and makes the runs go faster. In the lower levels, classes can need on armor that is lower than theirs. So hunters can need on cloth -- that they'll never use. And some of them will. If you are there at the beginning of the run, suggest a Need Everything loot system for speed purposes. If you aren't there at the beginning or forget to communicate, I recommend you make a quick gear comparison by holding down the shift key while you mouseover the dropped loot. Then Need it if it's an upgrade. Need every pattern for your professions too. If something drops that you can use and will equip right away, but you don't win it, just ask if the winner will wear it. You can trade dropped gear from the current instance and conjured items across realms but nothing else. To reduce drama, equip new gear right away.
Don't degroup from good groups
If you still have time in your play session and your group was a good one, don't degroup at the end. Speak up quickly and suggest requeueing together. You'll still get the satchel at the end because the group began as random. The only problem is when one of you outlevels the others, then the leader won't be able to requeue until the higher level leaves.
If you are stuck in a level where you get the same dungeon over and over along with everyone else you are PUGging with then the runs are going to be speedy blurs. But if it's your first time in a dungeon on your current character, announce that at the beginning. It shouldn't be difficult to find others who are willing to take it slow-ish to really enjoy the experience. My PUG and I had a blast in Deadmines even though we wiped a couple times and we got lost running back. A couple impatient DPSers dropped group because we weren't speedy enough, but they were instantly replaced. (That's another reason being a tank or healer is best.)
When you spend an evening playing with the Dungeon Finder Tool, you are going to find that the leveling will be fast and furious. You may even want to macro a "Grats" emote for all the groupies who are dinging with you.
Also, the Gearscore/Recount elitism that happens at endgame is nonexistent while leveling, making the whole experience more relaxing. (It seems that many of you have found that the elitism is still going on in leveling instances. How disappointing.) And at the end of the session, you'll find your more advanced character sporting sweet new armor, weapons and a bulging pocket of gold. Now go forth and dungeon crawl!