The fastest way to gain experience depends on what you're best at.
For instance, if you can consistently place close to the top of the leaderboards in a multiplayer game mode, that's probably going to provide the best experience gains per unit time spent. Don't play a free-for-all game type if you're more of a team player, and vice versa.
Likewise, if you're terrible at multiplayer, you're almost certainly going to get better experience bonuses from Spartan Ops. If you're going to play Spartan Ops for XP, go for the Matchmaking version. You can only play the current episode (for now, at least) and you must play Heroic. However, episodes go by quite fast, there's no friendly fire, and you'll tend to end up with ~4-6k experience per op. I find this to trump any XP per time invested in any other mode. If you've got double XP tokens, this is the place to use them.
When it comes to "War Games" style multiplayer matchmaking, I find personally that the "unusual" team game types tend to net me more experience. Anything with a Wraith, a Warthog (if I have a good teammate), or an abundance of power weapons and I'll do well. If you don't play with a coordinated party, these might not be the best for you. I have friends who insist that team slayer is the fastest way (or SWAT, whenever that becomes a playlist...). For me, though, I'll lose a BR or DMR fight often enough that I tend to end up with a K/D of less than 1.
Do note that you'll gain an good chunk of experience just for finishing out a round in online matchmaking, so don't quit early if you can avoid it, even if the game is going poorly. This bonus is by design, to give incentives to people for not quitting early.
Another thing to pay attention to is the periodic challenges. You can see these by pressing Start on most of the menu screens (this menu is called the "Spartan Hub"). Select the Career pane, and then the Challenges button. There's big bonuses on offer for varying time periods in various game modes.
Keep these in mind when you're deciding what to play - the playlist and difficulty matter for some, but for others you can set the game to easy and play on a level/game type you know well and walk away with a sizeable gain in a short amount of time. Whenever I start to play, I'll check out these challenges and try to knock out the fast ones before I do whatever I'm really planning to do.
In public multiplayer, based on the experience I've had so far - the first 3 seem to be Legendary difficulty and the last 2 Heroic. Every time I play I get this same structure.
The only way to select difficulty is to start a private Spartan Ops match and you can choose the difficulty, but it doesn't appear to me that you can make it public, so you either have to go it alone (Might suit some, but I don't fancy it!) or have enough available friends.
Quite a few achievements in this game are frustrating to get because the game leans you away from the goal!
Best Answer
While they closed one of the Reach Wraith-stealing tricks, they apparently didn't close the other.
To do this, you need to:
You can do this in Campaign and Spartan Ops, it works fine in either mode.
With regards to the related achievement (Mortardom), Here's a video guide, courtesy of IGN:
There are a lot of Wraiths in the section of the level once you've met the Composer, although at that point there will be friendly Rhinos competing for kills. The easy way to deal with this is to ignore the unoccupied Rhino, and ask the Marine in the other Rhino to hop out and take his. The rocket Mongoose can be somewhat of a pain, so you may wish to arrange for an "accident" (or just take his rocket launcher). Nuke the ghosts and then you can take on the Wraith. It's a bit tricky to manage all of this and get a plasma pistol, so you might consider bringing one from earlier in the mission.
You can also (with some effort and potentially some careful Wraith damage) get the first Wraith (where you grab the first target locator from the downed Pelican) to fit in the Mammoth and drag it around to the various objectives, if you are so inclined. Just punch the little "wingtips" on either side until they snap off. This takes a good number of punches. Be careful about wrangling the Wraith into the Mammoth, especially once the Mammoth starts to move. The Wraith has a tendency to disobey the laws of physics and clip through the Mammoth. It will also despawn when you get to the point where you have to go it alone on foot.
Keep in mind that the Mammoth has sniper rifles on the outer deck, which can make taking gunners down easier. A BR/DMR/Carbine works OK, as long as you're not getting shot at too much from the Wraith or any other unfriendlies in the area.
During my solo Legendary run, I also ran into a glitch when the "unidentified Covenant cruiser" appears where Wraiths and Ghosts kept spawning without moving the mission progress forward. This was an easy place to get Wraith kills, although I'm not particularly certain how I caused this glitch or how I ended it once it had started.
I took the Gauss Warthog that shows up in this area and parked it in the Mammoth, and was getting in and out of it repeatedly, which seemed to cause checkpoints often. Phantoms kept dropping Ghosts and Wraiths. I've gotten the game to do this twice now, so it may be worth a shot if you're having issues getting the kills you need.