My classic difficulty game had been going well. As soon as the first terror mission showed up, the overall difficulty spiked, but against all odds, my ground tactics were able to win out. After several successful missions, I managed to get my country ratings nice and low.

But suddenly my game is fucked. First I had to let a large UFO escape since 2 of my interceptors couldn't even scratch it. Terror levels went up. Then less than 24 hours later a very large UFO showed up and all I could do was throw up my hands and let it destroy my satellite. Suddenly, with only 1 day of game time, my situation room is all orange and red.

I'm not even that far into the game. What on earth could I have done to prevent this? Of all the research I've done, none of it yielded much benefit to my interceptors (just targeting). Is there a research path I should have taken? It just feels so out of my control.

Upgrade your ship's weaponry or, even better, build a Firestorm.

Firestorm with a Plasma Cannon can shoot down literally anything.

@Terramagi: How exactly? None of my research thus far has yielded any upgrades.

Don't remember exactly which ones it is, but a few of the early autopsies makes you able to make aircraft power-ups.

@mason: Experimental warfare research I believe allows for the Phoenix cannon Interceptor weapon. Heavy beam weapons allow for the Laser Cannon, and researching a UFO power source allows to to then research the Firestorm (you might need a certain number of Nav Computers and Power Sources first). There are more upgrades from there I'm sure, but that's as far as I've gotten.

The consumable ship items seem to be key for the larger UFOs. The floater autopsy gives you the defense matrix which temporary boosts your fighter's dodge rate, and a sectoid autopsy gets you the targeting boost. I was able to shoot down a large UFO (barely, I got very lucky) with an interceptor equipped with a laser cannon and using one targeting boost.

Reporting in on being at the doorstep of the final countdown on Classic Ironman. I can confirm that as long as you have the latest weapon upgrades for the interceptors you will be fine. Later on in the game when you try and shoot down battleships or terror ships it might be a little luck-based if you only use one Firestorm with a Plasma cannon, but the consumable items for interceptor combat is not required, it helps a lot though.

Edit: I grossly recommend capturing aliens even when you have already interrogated them. Plasma weapons are expensive and getting the research to use plasma weaponry makes the game just a step above a breeze.