1. Do lots of quests! It is possible to level up by just grinding, but doing quests makes leveling up go by much faster. It makes things more interesting, and the quest rewards and experience gain is nothing to sneeze at.
2. Don't bother reading the quest text. If you don't care about reading all the story text, just skip through and watch your quest tracker for the important notes you require. Usually you have to kill a certain monster, talk to a certain person, etc. There isn't really much else you need to know.
3. If you aren't reading the quest text, you might miss some quests. The good thing is, when you are headed off to do quest A, just kill one of each type of monster that you see along the way. Chances are you will already have one or two quests for that type of monster and you can kill two birds with one stone.
4. Do monster play and get destiny points! What is cool is that you can do monster play and get a ton of destiny points much faster than you can with your "good" character. You can use these destiny points for temporary buffs, or more importantly, for increased experience. Of course, the experience buff makes power leveling much faster for LOTRO. There is a limit to how many times you can do this. It can only be used once every 24 hours, and it will save up to 5 charges for you at any given time. This means that you should keep an eye on it and buy the experience buff whenever you can. If you let it fill up to 5 charges, you will be wasting free experience.
5. Don't be too worried about group quests. That is, don't spend a ton of time trying to get a group together. Usually the experience is horrible killing elites with a group, so don't count on leveling up fast. Of course the group quests sometimes have chains leading to future quests or great rewards, so don't ignore them completely - maybe wait until they are light blue or green so they will be easier to get over with.
6. Take up a craft. If you want to make sure that you are never short on gold, you might want to become an explorer so you can sell wood, leather or metal that you find during your travels. Typically you will want to focus on the highest tier of items available to you, and ignore the lowest level stuff.
7. Related to 6, make sure to visit the auction hall to upgrade your gear every couple of levels or so. Leveling is going to be alot easier with a fancy new weapon and armor.
8. Make sure to pick up some rations to help reduce your downtime while leveling.
9. Make sure to click on the stables as you go by - eventually you are going to have enough money to actually pay for it, and it will save you alot of travel time in the long run.