There are many different things you can do to level up fast in SWTOR. Reaching level cap in any MMORPG can feel like a grind and get tedious at times. If you use all of these tips, you can level up once or twice each day just by playing for a couple of hours. Leveling up in Star Wars: The Old Republic is much easier if you have friends to help. You will want to join a guild or play with real life friends to easily complete missions without the worry of constantly dying.
Once you have found a few people to level up with, start doing missions. Missions are the quests of SWTOR and give huge amounts of experience. Be sure to pick up all of the missions in each area before you head out to complete them. For the most part, all of the missions should be in the same general area so you can knock them all out at once. Most of these missions will have bonus objectives that will also give you huge amounts of experience. If completing missions isn't the way you want to level up your character, be sure to at least complete your character's class specific quests so you can unlock your ship, companions and legacy. Once you have unlocked your ship, you will have access to space battle missions. Some of these missions are repeatable and you can complete them once daily.
PvP is a great way to level up in SWTOR. Each PvP Warzone completion will give you a large amount of experience and credits. You will also earn Warzone Commendations that will give you access to PvP gear that you can use to upgrade your current gear every few levels. PvP with your friends and guild members for a better chance at winning games. Winning gives more experience than losing. The faster you win games, the faster you level up. You will also find a daily PvP quest in your faction's fleet that gives a lot of experience. Be sure to complete this mission daily for the most experience possible.
Flashpoints are the "dungeons" in Star Wars: The Old Republic. These dungeons are full of elite mobs that will give you great Xp. Group up with your friends, guild members or other players looking for flashpoint groups around your level. Be sure to finish all of the bonus missions available in each of the flashpoints for the most experience possible. You will also find daily flashpoint missions on almost every planet. Complete these each day to gain huge amounts of experience.
Be sure to save credits as you level through SWTOR. You will need a lot of money to buy mount training and skills at higher levels. Leveling up crew skills is a good idea but do not blow all of your money sending your companions out to do the work for you. If you start to feel burnt out from leveling, try a different method. I did PvP to level up most of the way in SWTOR and got burnt out around level 30. Once I started questing, it was like a whole new game.