There seem to be a number of tier-2 weapons, for example mass drivers, gauss, sonic, various kinds of laser, later plasma. It would be useful to hear thoughts on which ones are actually useful and which ones less so.
Mass drivers are poor-man (in black)'s gauss, lacking auto-fire and having less damage. They also weigh a little less. Yet, MD, gauss and lasers kinda all useful, all found relatively at the same time, and I'm not sure one can't do other's jobs.
Still, lasers feel like universal cure so that's what I usually go with. They're lightweight, accurate, have decent damage, usually good against robots, and limited ammo is not really an issue if you ration it properly. Gauss cannon works wonders on crafts though.
Sonic weapons can be found earlier, around 98. No auto, meh accuracy, damage falloff with range, but GOOD damage, thus good in close combat. In particular, sonic shotgun is small (1x2 in size) and powerful - best breaching sidearm. Also, sonics are handy underwater.
Plasma is still plasma, but there are some endgame targets resistant to it, so it needs to be mixed with something like lasers.
TLDR, commonly you can get through the game with just common firearms and lasers, maybe sonics for underwater missions.
How much is alien PSI going to be an issue?
In my experience, none whatsoever. I had my dogs/rats controlled by sectoids, though. Also there is one or two late game missions with strong alien psionics. Can't say they
need psi-proof team.
PStrength, PSkill, bravery, and current morale is used to defend against psi-attacks. So agents with low PStrength 30-60, but maxed PSkill (50) & Bravery (100 + commendations, 120-170) can reliably withstand 95% psi-attacks in the game even on superhuman. I deployed troops with less 30 PStrength veterans against Ethereals without issue, but not that many times, so can't really vouch for it.
But still, it makes sense to recruit rookies with 80+ PStrength. Offensive psionics should only be 80-100 PStrength hybrids. Humans psionics are usable, but meh even at 100 PST until late game (olympianism).
There is
rare mission in the game that allows you to raise PStrength by 7.
which agents should you eventually transform to, for example, spartanism
Olympianism is for human psionics, and proteanism is for meaty assaults. Both not strictly necessary. Both mess with psi-def, boosting it a little in case of the latter.
Kyberos is currently standard transformation. It raises stat caps for firing, health, reactions, etc., by about 20.
Spartans seems to be in the works now. It offers 40 TUs to boot in addition to 20~ish bonuses around, and no PSI resistance issues. I guess it'll be new meta but we'll see, I'm not sure these bonuses are applied as-is.
And finally, what are the key things you should have when the invasion begins (e.g. radars? interceptors, are the early missiles sufficient? etc.)
1-2 large radars per base, and 4-6 bases is enough to detect most UFOs. Early, you can just tail UFOs with your transport until they land, like good old times. But there are some hunter-killer UFOs that pretend being docile... Ravens\Thunderstorms (with Heavy Stingrays) are first adult interceptors, and should be used in squads of 2-3. Migs\Interceptors can be ignored.
In battlescape, at minimum, dogs and smoke can beat most early aliens without much losses.
Tactical grenade launchers, lasers / blops snipers help clearing outside of UFOs.
Throwing axes are good to one-shot both soft and hard targets like hybrid drones or medium-armored MIBs. EMP grenades and mines are good to have against robots/heavy mibs/tanks/sectopods.
Armor is also not really necessary for beating early aliens, but raises comfort quite a bit. Plasma rifle one-shot kills of 70hp agents are unlikely even against personal armor (or trit vest if taken strictly upfront). Cyber armor is slightly better and offers decent overall protection. Power/flying suit is only trumped by Shock in plasma def. Assault/Stormtrooper trades some defence for mobility and kicking butts in melee. Also, assault moves very well underwater, and stormtrooper sees very well underwater. But you usually complete underwater before both...
Magma power armor is bad against plasma, but good against kinetic. Which is kinda sad, because it's earned through beating reptoids with kinetic...