If somebody is weak and they have a bunch of cards it might be a worthwhile gamble to take them out, even if you end up stretching yourself thin. Partly to keep your opponents from getting them but also for massive wad of armies you get if you get to turn in two sets.
Screw continents.
I used to think I was a pretty good Risk player; I could beat most people I knew most of the time. Then, in college, I idly decided to play some Risk with a guy at work - incidentally, one of the smartest people I’ve ever met.
I got my ass whooped.
Over and over.
Consistently.
I got my ass whooped even after being dealt the entire continent of Australia plus Siam.
Why? Because I was following the conventional wisdom. Defend your borders; play slowly; try to get a small continent that you can defend. Australia, South America - those are the keys to most peoples’ Risk strategy.
In reality, continental bonuses are nice but they are not vital. Especially early in the game. If you win two extra troops per turn for having a continent, but spend more than that defending it, you’re losing troops.
In the early game, concentrate on accumulating land. Think of it this way: six extra territories is like owning your own private Australia, and you can set them wherever you want. Lose one? Take a different one. It’s far easier to do without anyone really noticing - you count your own territories, but no one else’s.
There is no part of the Risk board that’s vital; there isn’t even a part that’s particularly better than the rest of the board. Whether or not you allocate your troops randomly (this is a matter of house rules) try not to abandon too many of them. In the early game, even a very spread out army is fairly defensible because most people have very few troops. The conventional wisdom would say to consolidate and abandon ‘unproductive’ areas, but I learned that it’s far better to take advantage of the army you get for every three territories than to try to hang onto a tiny pocket.
If you’re more spread out, you’re also less vulnerable. A set of cards, once the game gets going, can usually defeat any border, no matter how heavily defended. You’re counting on your troop bonus from Australia or Africa? Gone. But accumulate enough sheer area of the board? Well, they can’t take that away anywhere near as easily. No part of your territory is key to your survival, so you can deal with any attack.
Personally, I try to prevent anyone else from getting an early continental bonus. But DO NOT get dragged into an early war; if someone’s going for Australia, slow 'em up but don’t try to fight 'em for it. Concentrate on taking a couple other places instead - no one will care if you have the Ukraine, and Kamchatka, and the Eastern US, and Central America, and Madagascar, and South Africa. Between six extra territories? Two armies a turn. Everyone wants a piece of you when you got Australia. Shun the spotlight. Australia is great if the players are mediocre, but if you can rise above conventional strategy, you’ll eat those didgeridoo-playing suckers for lunch.
Play offensively. The “Sit tight in Argentina” strategy that’s so common is a defensive strategy. Defensive strategies are attractive, but they don’t stand up once the card bonuses hit twenty or so. After that, a sharp blow can knock out any continent, and it’ll happen to you. Don’t spread yourself too thin, but don’t be afraid to spread out. Take stuff from other people. Try not to target one player because they’ll hate you. If your friends play politically, start alliances. Agree with a friend that neither side will fortify one long border. Offer someone a country they want in exchange for one you want.
Break alliances when it suits you. Or develop a reputation for keeping your word - either one can be highly useful.
Build intercontinental territories - areas that span multiple continents, forgoing continental bonuses but denying them to others; you can make them as defensible as you please and no one will resent you for it.
Remember, sheer superiority in territorial numbers defeats just about any other aspect of the game.