Please, the shamblers aren't that bad. Fire, or just a hail of lead will kill them. You can kill them early with just a crap ton of colt 45 fire, since they have almost no armor, and they are vulnerable to flareguns too
As for UFO's, well.. ...
...It is indeed possible early on, but you MUST think very carefully about how you do it. Sectoids are weak and fragile, so it comes down to more getting the jump on them, and using terrain properly than having the right gear. I beat them with just shotguns, kevlar, and hunting rifles, but the biggest challenge is 2 things that early
1: Getting a UFO landing is almost pure luck early on. Lets face it : trying to chase a UFO on a commercial flight is a silly idea, but if you get lucky, you can totally do it, if it lands. Always at least try if its close, worst you will do is waste a few ingame hours.
2: you will have 0 survivability. Kevlar is a really crappy defense against ANY alien weapon, even a crummy alien laser or pistol. Lasers will melt right through the kevlar plate generally, and kill you instantly : Generally you 1 shot sectoids also, but this doesn't make it less of a problem. Also, you get 4 people THAT early, so your margin for error is EXTRMELY low. This said, capturing a UFO THAT early is a huge boon : selling off the UFO parts, corpses, weapons, and the points alone for capturing a UFO (even with 3 / 4 agents killed) is a HUGE point gain that early, enough to turn a failure of a month into a decent month
Overall, just be mindful of what your gears capability is compared to what you are facing, and be aware of how vulnerable you are. Sectoids are fragile enough you don't need to worry about shooting multiple times, unless you get a VERY bad damage roll, so its mostly making the shots count, and not getting shot yourself.
Also, don't be afraid to attempt a night raid on a UFO : with flares, flashlights, and keen planing, the dark can actually help you : just make sure you put your flashlight dudes in cover, leave scouts in the dark, and throw flares into areas you are going to scout. But I admit, raiding a UFO before you get MP5s at least is a very bad idea (one I did anyway)