I'm current mid way through a play of a long war game and thought I'd offer a few bits of advice I've determined from experience and researching within the wiki for my own knowledge.
First of all, gaining an upper hand in aerial combat is going to take a long time. The mod intentionally makes the air game a constant lose battle. As always, if you rush to these improvements first you'll likely suffer in the tactical game, so a careful balance is required.
There's a few ways to improve your effectiveness in dealing with UFOs:
Pick your Fights
Don't try and shoot down large, or even medium, sized craft until you've improved your air force. A lot. You really just have to let them go at the start.
Even the small UFOs will require multiple stock interceptors to shoot it down, so don't go into a fight unless you're happy to have damaged interceptors for a while.
Scare them Away
You don't always need to shoot down a UFO to (possibly) prevent it from completing it's objectives, sometimes just damaging it is enough. This allows you to still deflect some missions at the cost of less down time for your interceptors.
For example, a UFO that's hunting satellites will have a reduced chance of success if damaged, and will fail if it's health has dropped below 50%.
Once you have completed a tactical mission against a type of UFO - either after shooting it down or because it has landed - you will have the option of performing a UFO Analysis research on that type of craft. Completing this reasearch will give a damage bonus to all inteceptor attacks (with all weapons) against that type of UFO in the future.
As you research better weapons you can construct equivalents to be mounted on to your interceptors. The path through weapon research if fairly obvious so I won't repeat it here.
Also, in the early game when you're still using the default weapons do note that the armour piercing Stingray Missiles work best against the armoured Fighter small craft, while the Avalanche Missiles work best against the Scout small craft - so it can be worth swapping the default weapons around a bit.
As with the vanilla game, you can use Tracking, Aim and Dodge modules during combat to temporarily increase combat time, interceptor accuracy and interceptor change of dodging shots.
All of these require the Aircraft Boosters foundry project to have been completed - this is available after completing the Alien Materials research, which is available immediately.
Tracking (Aircraft Booster) requires no additional research.
Aim (Uplink Targeting) requires the Floater Autopsy research.
Dodge (Defense Matrix) requires the Cyberdisc Autopsy research.
There are several Foundry projects available that provide perminant upgrades to all interceptors - more health, better accuracy, dodging, etc - that can be used to improve your chances in the air.
Many of these are unlocked by the Advanced Aerospace Concepts research project, which will requires Experimental Warfare, Alien Weaponry and Alien Materials to start. The Seeker Autopsy and Alien Propulsion projects also unlock other upgrades.
There's too much information to repeat here, so go have a look at the UFOpaedia's Foundary (Long War) page and scroll down to Aerospace Improvements for a full breakdown.