In the first instance, report to the server owner. They’ll usually have forums you can post on, or an email or something linked through a steam profile.
Otherwise you can report through Steam’s support system, I think, but I haven’t ever done it so I’m not sure how.
You can’t report hackers.
I got TF2 with the Orange Box ages ago but never played it because A) I suck and online FPSes and 2) The game’s been around so long I figured all the players would be uber-leet and no fun to play with.
So, when it went Free To Play, I figured the servers would be full of people who are worse than I am 
and decided to give it a go… I’ve really been enjoying it so far!
I’ve found I’m most effective as either a Sniper or a Pyro, but I’m still having a lot of trouble dealing with Spies- other players seem to instantly materialise behind my avatar, backstab it, and instantly vanish, yet whenever I play as a Spy, no matter how cloaked I am or how cleverly disguised, opposing players still seem to unerringly zero in on my character and then I’m looking at the “Waiting For Respawn” view again.
So far I’ve found the Demoman and Scout classes the most challenging, FWIW.
It’s possible that you’re both just following the same line of thought - you think it’s a good place for a spy to hide so you stand there, he thinks it’s a good place for a spy to hide so he shoots there a few times to make sure.
QFT. The corner at the base of the stairs in the sewer in 2Fort is a great example of this. Every. Goddamn. Time. there’s a spy there.
Possibly, but there’s still times when I’ve been shot whilst moving by someone who couldn’t possibly have known I was there. It seems to be related to an issue I have as an engineer… no-one’s around, I’ll check with the wrench to make sure no hidden spies, then I’ll build a sentry gun and suddenly a spy appears, saps it, backstabs my engineer, and vanishes again.
Like I said, I like the fact that F2P has brought in lots of players who aren’t super-awesome since it means that average players can enjoy the game too without being made to feel like a nuisance or inconvenience for the most part…
To be honest, TF2 has always been pretty forgiving of noobishness. Especially on the Aussie servers, IME. It’s not as twitchy as games like Counter Strike, and the cartoonish nature seems to appeal to a more forgiving class of player. The influx of ubernoobs hasn’t changed the tenor of the game much, it just means there’s more players around.
The thing about spies and spychecking is that good spies will wait on the periphery until the spycheck is done, and THEN dash in with the sap-n-stab combo. And a lot of players will spycheck constantly as a matter of course. As I’ve mentioned before, I play Heavy pretty exclusively. So I hang near ammo sources and shoot everything and everyone that seems suspicious, and occasionally the walls for good measure. So I’ll randomly spycheck, see a flash of red or blue and try to anticipate where the bugger is leading, so I can blow him off the map with a full fire.
Once you’re accustomed to the game, you can tell a spy simply by how they move around the map. They stand out plain as day to an experienced player and, consequently, experienced spies adapt to that and focus on not being seen in the first place. Disguises are only there to not instantly flash as hostile at glance, to buy the split second needed to strike a distracted target. You can take the disguises away entirely and it often doesn’t significantly change anything (see the Eternal Reward knife). “Spychecking” as new players focus on is completely a non-issue - only greedy or newbie spies will give you that much opportunity to shoot them in the first place.
You counter spies by having good situational awareness in general. There’s no gimmick to it, just have a good feel for what’s happening around you and don’t get fixated on any one thing. It’s an essential ingredient for performance in every aspect of the game for every class.
That said, even the masters of the game generally don’t stop the really good spies from killing them. Most don’t spend much effort trying, as newbie spies are easy meat and the good ones will get you anyway: everyone has moments when they’re vulnerable. What they do is stop the spies from killing everyone else around them. The pyro who runs around burning teammates to no purpose isn’t a threat, he’s obvious, predictable and short-ranged. It’s the soldier who instantly spots an attack from across the room and lands a rocket just before someone else gets stabbed - he’s the one spies fear. It’s very much a teamwork thing - a spy killing streak is half due to the spy’s actions and half due to the other team allowing it to happen.
It’s fun doing that as a sniper, actually… see the coloured flash, work out where it’s going, lead the target, and hello “Shoot The Breeze” Achievement. 
I have to say I love the silly names the Achievements have- I laughed heartily when, playing as a Spy, I was able to dominate a (clearly newer then I was) Sniper and got “The Melbourne Supremacy” (a pun on “The Bourne Supremacy”) Achievement.
And yeah, the Aussie servers do seem a bit more laid back, which is nice- I play games for fun, not to be shouted at for “Doing it wrong” by people who clearly have too much spare time. 
I don’t know if this has been said already, but airblasting an enemy off of a cliff does NOT count as a suicide for them. It used to, but Valve changed it so that it now counts as a kill for the pyro.
Sniper is my specialty. 
I use the basic sniper gun, the Jarate, and the Tribalman’s shiv.
To me, no other rifle can compare to the original sniper rifle. No drawbacks to it, just charge your bullet and aim for an enemy. Headshots get critical hits, and it’s entertaining to see how high the amount of damage can go when you enable the visible damage counter.
Jarate is perfect for both offense and defense. Offensively, you can throw it into a group of enemies and it will coat multiple at a time. It gives mini-crits, meaning you can step back and allow your team to jump in and waste them away more quickly and you get points as kill assists for any jarate’d players that are killed. If you sense that there’s a spy around, you can throw the jarate near him. Even if he is invisible, he’ll be covered in a dripping pee-ish hue, allowing you and your teammates to see him well enough to kill him.
Defensively it’s great for putting out pyro fires that are either on you or your teammates.
The only downside to Jarate is waiting for the thing to fill up again. I definitely don’t mind losing my second gun to this, though!
The Tribalman’s shiv isn’t as damaging as the kukri but it has a bleed affect, meaning that after the initial hit the enemy will continue to bleed out for some time. It is only stopped after the hit player finds a health kit or after the effect wears off after six seconds. This is good for defending yourself, you’d be surprised how many people run away after getting hit with one! The delay affect is nice too, especially if an enemy runs away after a couple of hits only to die shortly after from the bleed effect. It’s also great for spies- if you suspect there is a spy among the teammates or invisible somewhere, whack the said teammate/area. If blood starts spurting, you’ve caught yourself a spy!
With my layout, you can’t exactly run into the middle of the battle (with the exception of throwing the jarate into a group of enemies, but you have to run out shortly after if you want to live.) That has never been my method with sniper, anyway- I’ve always imagined him as the guy who quietly picks off the enemy from afar. So far, so good!
Why would you want to bleed someone instead of killing them? The Bushwacka is the OP sniper melee weapon, especially if you’re going to carry Jarate.
One hit with the standard Kukri will not kill a spy anyway, while one hit with the Shiv will always identify him. I’m not a particular fan of bleeding as a trade-off for damage, but the reasoning is sound.
Being able to track spies just isn’t a thing that you need to do. What you need to do is one-hit the soldiers and demomen who jump over to you. That’s the way bigger threat. You don’t give up so much against many classes for a slight boost against one that you aren’t going to be fighting anyway.
I don’t care for any weapons that lower your resistance to fire damage. Sniper’s tough enough as it is when you got a pyro running after you. Bleeding also helps identify the enemy to more seasoned teammates so if for some reason you can’t finish them off, they can finish the chase.
Chances are as a sniper if a pyro gets to you you’re gonna die anyway, so it’s not a huge downside. Took me forever to get a bushwacka to drop but I use it now - quite good with jarate.
Bleeding helps identify cloaked spies. You’ll still an invisible space bleeding out. So it’s nice there as well.
I stumbled across a couple of… odd servers. One where you’re turned into various objects and you play hide and seek with some pyros. Another where opponents don’t take damage but the ground does (it crumbles if you shoot it a few times, revealing liquid magma below) - so the goal is to run around and shoot the ground out from under the other team.
What else is out there? Anything really wacky or cool we should check out?
There’s at least one Australian server hosting “Surf” maps, where you get around not just by running (as usual), but by “surfing” along ramps to various locations. They take a bit of getting used to, but once you’ve worked it out, they’re surprisingly fun and chaotic, even if you’re new.
One of the servers I used to play on (Best GN, but I don’t think they exist any more, I haven’t seen any BestGN servers around in yonks) would occasionally turn on “Party mode” where damage dealt resulted in balloons and streamers scattering everywhere, and when you got gibbed by an explosive you’d disappear in a shower of presents. That was lulzy, if not really game-changing.
Oh, and I love 24 hour rounds in 2Fort. I know, I know. Everyone hates 2Fort. But I really enjoy it
Actually, I do too. But evidently on SDMB if you like 2Fort you are not a Very Serious Player.