My personal experience is limited here, but my thoughts are:
1) Don't buy an extender for just the lenses you have today. Think about what will be in your bag down the road, and with that in mind, get the most compatible tool you can.
2) I own a 2x III and only have one lens (the great 70-200 F/2.8L IS II) that I use it on. Though it's somewhat less sharp than the native glass, you know the argument -- less weight/size to carry, less risk to smuggle into events that you shouldn't bring large glass to (read: sporting events), less heavy/expensive glass not getting used, and far less cost. For me, as an enthusiast (i.e. not a pro), this is an easy decision for the aforementioned reasons, and I love my extender as a result.
3) AF speed with an extender does suffer a bit, so sports may be problematic depending on what you shoot. I've shot baseball with it very well, but I've only been framed on pitchers or batters (which are easy targets) -- hockey, basketball, etc. would likely be a nightmare with one.
4) If you are a pixel peeper, let Bryan Carnathan show you the way...http://www.the-digital-picture.com/Reviews/Canon-Extender-EF-2x-III-Review.aspx
As always, the good Mr. Carnathan has nice mouseover comparisons of the different extenders. Look about 40% of the way down the page for the sharpness comparison.
Also, he has controlled lab shots with/without extenders here:http://www.the-digital-picture.com/Reviews/ISO-12233-Sample-Crops.aspx?Lens=687&Camera=453&Sample=0&FLI=3&API=3&LensComp=687&CameraComp=453&SampleComp=0&FLIComp=7&APIComp=2
Feel free to noodle with apertures and extenders (just change focal length, the 1.4 or 2.0 should be obvious based on seeing 280 or 400 on a 70-200 lens). Then mouseover left to right on the graphic to see each. Super useful!
Just my two bits.