Don't worry about your backswing so much. If you want to increase distance try turning your hips and shoulders faster. When doing this the tendency it to hit fade or slice so you will need to close the clubface earlier in the downswing as well.

Swing easy.  A friend of mine that just crushes the ball always says "Swing like a lady, fly 280"  (no offense ladies).  Every time I try to kill it I end up hitting the big old slice, or misshitting the ball.  Focusing on hitting the sweet spot will really help you even more than clubhead speed.

There's also a drill where you hold the driver by the head and take a swing at an imaginary ball.  Focus on rotation of your shoulders, and listen for the wooosh sound the grip makes as it swings by.  It should wooosh after 'impact'.  That way you know you're getting the most speed right at impact.

ok there was a post above mine about sholder turn and closeing the club face. Scratch that whole advice, it was kinda horrible.  he is just promoting bad swing mecanics. number one how old are you and what kind of driver do you use. what are the specs is it a r flex s,x?  whats the loft whats the kick point in the shaft and also whats the shaft weight.  all this dose matter. next whats your swing like? how far back to you take the club. how are you holding it at the top of your swing. there are lots and lots of diffrent things that affect how to gain distace if you only hit 218 yards. do you already have a slice?  sorry if this isnt the answer you want but your discription of your problem is very  vage you just say i want to hit further  but with out knowing what your really doing wrong i cant help you know.  here is alittle advice go to youtube good jim mclean 8 step swing work on that for alittle bit and tell me if that is helping you hit farther and straighter.  other than that i have no idea how to help you with out more info. sorry but maybe seeing your local pga pro is exactly what you need.

Thanks will try Swinging down through faster. Herman

4 RIGHT HANDED PLAYERS.  IT AMOUNTS TO MORE BEND IN THE BOTTOM (RT) WRIST.  THE RIGHT PALM SHOULD BE BEHIND THE SHAFT AT ADDRESS.  THE RT PALM DRIVES THE SHAFT FORWARD.   THAT COMBINED WITH BODY ROTATION SHOULD LET YOU KEEP THE SHAFT INSIDE THE BALL FOR GREATER LAG.  TRY IT YOU'LL LIKE IT.

sas,

First are you a junior golfer? If yes, be patient and keep on playing, and practicing ... your distance will come (with all clubs).

Make sure you frequent the range and work on basic swing fundamentals along with developing a smooth tempo. Don't be afraid to have a PGA pro look at your swing. Also use the internet, and watch how the PGA tour players hit their shots. Now as your swing improves, so will your club head speed—as your overall swing will become more efficient. Additionally, assuming you are a junior golfer, each year you'll have more strength, which will also add distance to your game ... but again, I can't reinforce practicing solid swing fundamentals. When I was a junior, I gained over 50-yards with my driver between the ages of 15 and 16. 

One last observation:
If your 3-wood distance is the same as your driver's, there's a chance that your driver doesn't fit you correctly. The driver can be the toughest club to hit in the bag and an incorrect fit will only compound this. Again consult a PGA pro, they will look at all aspects of the club: it's length, weight, shaft material, shaft flex, and its loft. You'll be amazed by how you well hit a driver that's either fitted or at least matches to your your swing type.

This is your first year playing golf. And you hit the ball 200 yards, you're doing just fine.

As I was once told, "don't worry about distance, it will come, smooth is better than hard." You've probably been told the same thing. If you have then you are doing like I did, not listening.

The more time you spend on the range, swinging correctly, the faster your brain can make the movements and keep it smooth.

BTW a full shoulder turn is essential to my swing, and if I don't have a little hip turn I pull 10-15*. There are 10 million ways to swing the club, and there will be different advice from different people, you need someone that knows your swing.