Just about.
As the others are saying, get some foam board and build a Flite Test Tiny Trainer from the free plans. You can build this as a glider and throw it around until you're confident with the controls, then add a motor to make a trainer - and then build the 'sport' wing and upgrade the battery to make it aerobatic.
Others are recommending the FS-i6 transmitter, which is a good choice as it'll be good when you get a few more models - but you can go a little cheaper if you don't mind only having one model. $32. If you stay in the hobby you'll definitely want something better before long though - but if you get serious you might want something even better than the i6. After a few years most people are using transmitters in the $100-$300 range.
Here's a 12amp ESC with a BEC $7 - many quadcopter ESCs don't have a BEC, which you need to power your servos.
Here's a pack of 5 servos for $11 so you have a spare
Two 2s 800mAh batteries for $20
Here's a $11 charger - this is another item you'll want to upgrade if you get into the hobby, so you can charge bigger batteries, or several at once.
That's $91, leaving you a bit for foam board, hot glue, music wire (for push rods) and postage.
These links are for reference - I've not tried this exact stuff, but it matches the requirements of the plane. With a bit of hunting you might find the same things even cheaper - especially as I ignored the postage.
I went with a Turnigy 2200mAh 3S 20C Lipo Pack for the controller with this charger and they are XT60
Could I use the same charger for the Quad packs? I have not pulled the trigger on the Turnigy nano-tech 1300mAh 3S 45/90 Lipo Pack yet.
I'm guessing the recommendation for the 1800mah 25-35C is to fly slower at first right? We would by flying FPV but not racing it so I think that's probably a good recommendation.
Since we are already using 2200mAh 3S 20C with an XT60 connector could we just use the same exact pack for the quad? It's rated at 20 constant, 30 burst and we could always use a spare for the controller if we go with a lighter battery with higher discharge for the quad later.
We aren't carrying a GoPro or anything so it would probably support the 2200.