Operation Flashpoint: Red River

The maximum level for all classes is 20. Each class has B-Mods that increase experience. The first B-Mod gives you an additional 10% experience and the second, unlocked once you reach rank 13, nets you an extra 20%. I suggest equipping those as it will make you rank up faster.

By playing the campaign once you should have reached level 20 if you stuck to the same class. You get experience for pretty much everything, from killing to healing. objectives seem to give you the largest amount of experience so I suggest doing all optional objectives and healing your teammates yourself rather then ordering them to do it.

If you want to get this done quickly you can play Fireteam Engagement missions from the main menu. Combat sweeps are quite fast and easy. CSAR missions are also not that hard if you know what path to follow and have someone to blow up the chopper for extra experience. One Combat sweep or CSAR mission can be done in 8 to 10 minutes.

Many thanks to Defrosted Jay for pointing out that you can check your experience while in game. In order to do so, pause the game and press . You will then see your Combat Experience tab. Here are the categories with experience gains I have observed so far;

Current Kills XP
How much XP you get depends on the type of kill. You can get a normal kill, headshot kill, vehicle kill, and explosive kill.

Current Objectives XP
Main objectives and optional objectives give you experience points. How much will, at least for Fireteam Engagement missions, only show as a sum at the end of the mission. During campaign you will receive stepwise XP every time you complete a major objective.

Current Teamwork XP
- Stopped bleeding on teammate 22XP
- Heal teammate (orange) 22XP
- Heal teammate (yellow) 22XP

Current XP Penalties
- Squad death (varying) Note: Reviving a team member while he is dying nullifies the penalty.
- Friendly Fire (-50XP)

Survival Bonus
As suggested by tweekfu in the comments, you receive a 10% bonus up to 30% for each mission you survived beforehand.

Actions with no observable XP
- Healing self
- Ordering teammate to stop your bleeding
- Repairing vehicle, self and teammates
- Orders of any kind

Many thanks to XboxFreak36 who confirmed that the experience needed to gain levels does indeed increase. At level one you require 4000 XP to reach level two. From level nineteen to twenty you would need 39000 XP. This might seem to be a lot but keeping in mind that you can earn about 30000 XP, depending on your skill of course, roughly every 10 minutes, you can get one class to its maximum level in a few hours.

Review 25 September 2014: Added button icons.

In addition to Epsilon Theta's XP guide, and if you find yourself struggling to get kills or complete objectives when playing as a certain class - try the mission 6 "Be careful what you wish for"

Which class you choose is pretty much irrelevant as you'll no doubt be picking up the M107 night ops sniper rifle at the start of the mission.

Play the mission as normal and continue through until you drop the JDAM, then continue onto the regroup rally point where the game saves and awards you XP up to that point.

If you spend a little time killing all enemies you should earn somewhere between 8500-10000xp in around 7-9 minutes (inc the 3 min cutscene at the start) depending how long you spend shooting enemies in the distance and if you've got the 20% xp multiplier mod equipped.

After you reach the rally point (Save the Kodak moment for later) and game saves, you can quit out and restart the mission.

This isn't the quickest way to earn xp but it's handy as you can do it on your own and you'll have the same weapon every time.

This is best achieved doing the fireteam engagements whtich are objective based and are quite short with the objectives giving ridiculous amounts of experience.

It can take up to a few hours to max out one class to lvl 20

combat sweep is best. 1 map in 10 mins can get at least 1 level per map

In Addition to what the above guides say which are spot on in what they say, ive found that doing mission 9 (vantage point) is a very quick level to get through even on solo obviously even quicker if you can get a good squad of people together. i went from level 1 to 20 pretty quickly just by doing combat sweeps combined with this level just to keep the grind a bit less tedious.