Speed
When you're trying to keep up with a monster, you have two options when it comes to Switch Axes: either sheathe your weapon and run or stay in Axe Mode. Because of the Switch Axe's reduced speed compared to Sword and Shield (really, every weapon is shown up by the SnS's mobility), you need understand more about monster pathing if you want to deal continuous damage.
Instead of following the monster wherever it goes, Axe users will need to anticipate where the monster is going. As you gain experience hunting, you'll learn what attacks certain monsters use and what position those attacks leave them in. As each monster telegraphs its attacks quite obviously, you can often figure out where a monster will be far before they get there.
But if you're still having trouble sticking to monsters, I'd recommend the Evade Extender armor skill (found as Evade Dist on individual pieces of armor). This armor skill allows to you cover more distance when you dodge roll or sidestep, very effective in allowing you to get to monsters without sheathing your weapon.
Combos
Combos are fairly straightforward when it comes to Switch Axes. Like a few weapons, the Switch Axe has what is termed an infinite combo. In fact, both modes of the Switch Axe have this infinite combo, although Sword Mode is limited by the phial gauge. This infinite combo is simply X, X, X, X, X... For Axe Mode, it is important that the first X is Forward + X and each subsequent X is just Neutral + X. Beginning with Forward + X means you start your combo faster and Neutral + X keeps you in one place while adding the overhead smash to your infinite combo, which is the highest damage move in the Switch Axe arsenal, barring the phial explosion.
One important caveat to Switch Axe combos: switching. While most weapons can easily deal their best damage simply whaling away at a monster, Axe users must understand the importance of switching and Attack Switches. Knowing what Mode you need to be in is crucial to maximizing your effectiveness as an Axe user. Typically, part breaking is left to Axe Mode as you have far more range with attacks in this mode, although easily hit parts can be broken more quickly by the Sword Mode's furious attack pace. Attack Switches are attacks that combine mode switching, allowing you to deal damage and change to either Axe Mode or Sword Mode. The Axe->Sword attack switch is Forward + X followed by R, and the Sword->Axe attack switch is Neutral + A followed by R.
Specials
Unless you count the Axe Mode's uppercut swing, which can indeed by activated by the special key on your Gamepad, as a special attack, the only special attack worth considering is the Sword Mode's phial explosion. That "helicopter attack" is almost completely useless and I wouldn't recommend using it at all. Each hit in that attack does a low amount damage and the stamina drain leaves you vulnerable. As each hit still depletes your Sharpness at the same rate, you'll also have to use Whetstones a lot sooner when you use this attack.
Using your phial explosion, or Discharge Burst, isn't what you would call obvious. Pressing X + A while in Sword Mode will start the Burst, but you need to continue to hit X to rev it up until it actually explodes. Using your Discharge Burst properly is key to becoming a good Axe user. The damage of the Discharge Burst is wildly different from your attacks, depending on what phial you use. Most, if not all, of the damage of the Burst is actually determined by your phial. Power Discharge Bursts deal nearly pure raw damage while Element Bursts unload a metric ton of element damage on your enemies.
Because of the amount of time it takes to detonate your Discharge Burst, it's best to dedicate an entire attack opportunity to it. Trying to fit it in at the end of a combo is really difficult and the monster is likely to move before you finish.
Best Answer
In the general sense, every weapon will give the same DPS at the same rough level of skill. If you are just as skilled with the Dual Swords as the Gunlance, your DPS is going to be roughly even. How much DPS you do depends on quite a few factors, and if all of these are equal, so will the DPS.
You can see the damage formula here. It's not just raw DPS you need to pay attention to, nor elemental. You need to ensure you are hitting the right area of the monster, using an element the monster is weak to, and you have to make sure you dodge, too. Monsters aren't just going to sit still and take it, either, so you can't just assume you can go all out and have perfect DPS.
In the end, DPS is somewhat of a meaningless stat, as it all evens out.