A couple of these I'm pulling straight from Richard Launius' House Rules (My favorite of which is Rule #12)
If you want an easier time of it:
Drawing a Mythos card every other turn is the way I've done it. It allows the players to wander around Arkham and have encounters they normally wouldn't take the time out to have.
You can also make traveling through gates (for those standing on them) during the encounter phase optional rather than mandatory.
Monsters standing on a location that contains a gate when that gate is closed are drawn through it, in addition to all those that would normally do so due to their dimensional symbol.
To make the game harder:
Have Investigators randomly dealt out to players or randomly dealt two and pick, etc.. My group usually hates playing spellcasters, and with good reason! They weren't particularly impressive before the later expansions and the spells use precious Sanity. This forces players to pick people they wouldn't normally play as and prevents games from being filled with powerful characters.
Place gates face down on the board. This makes it more of a surprise when you find yourself plummeting into R'yleh after jumping into an unguarded gate, or fighting through a horde to end up in the Dreamlands. It also makes it harder to come out of a gate that you didn't enter through (as they are facedown, you don't know if there is another exit on the board). When I do this, I usually allow the exiting Investigator to look at one facedown gate, and if it wasn't the one they were looking for, they get LiTaS and the gate stays facedown.
The Heralds and Guardians that made their debut in later expansions (King in Yellow and beyond) also help make the game harder (or easier with Guardians). Hastur paired with the King In Yellow Herald have made games so painful my group refuses to play with either one of them anymore.
Personally, I've never found it necessary to take away a character ability. None of them are that broken. Some are close, but there's always a drawback to them.
Additionally, (though I've never used them myself) there are difficulty modifying cards in the Black Goat of the Woods expansion. Two of them make the game easier, two of the harder. I think they do things like 'everybody starts blessed/cursed, more/less mythos per turn', etc.