Protoss Basics
There are many different protoss openings, though I recommend the following:

1. Send your initial probes to minerals (send them to separate mineral patches),
2. Keep building drones until 7 control,
3. When the 7th probe pops out you will have around 100 minerals, so build a pylon,
4. Build 2 more probes for total of 9,
5. With 9th probe warp in a gateway and explore the map.
6. Build 2 more probes and a pylon.

Countering Zerglings
Zergling rush can be devastating for a protoss player if not defended properly. By using the protoss basic build order you'll be able to get one zealot at most before the rush comes. So, you'll have to pull a few probes from the minerals to successfully counter the rush. You are likely to loose the zealot but very few/no probes. If you feel insecure about defending the rush with zealot and probes, use the following build order:

1. Build probes until 6 control,
2. Warp a pylon,
3. Build another probe,
4. As soon as the pylon finishes build forge,
5. When the forge warps in build 2 cannons (1st right out, 2nd when you get enough minerals).

With this build order you'll pretty much shut off any zergling rush but your development will be somewhat damaged because you're left without a gateway.

The Zealot Rush
Zealot, compared to the zergling and marine is a much more stronger unit, and you'll probably want to rush using them early on. The fact is, if you rush with 12 zealots, your opponent probably won't build up 48 zerglings or 24 marines for defence, but get some sunkens/bunkers and try getting more advanced units (in which case you can bypass them and hit the worker line).

1. Use the build order mentioned in the beginning,
2. Continue building a probe and a gateway,
3. After that keep cranking probes until you get the number that fits your minerals (3-2),
4. As each gateway warps in, start cranking zealots and keep up with pylons,
5. Get the 3rd gateway if you think you can support it.

After you get 8/12 zealots, attack and expand. At this point I'd also recommend to set up a forge for defence.

Offensive Cannoning
Offensive cannoning is more and more popular strat used over the Battle.NET. The thing here is to build a few cannons close to your opponent's base but out of his range of sight, then keep advancing more in to his base until you win. Obviously, this is a go-for-win strat, because no other units than probes should be built.

1. Use the build order I mentioned in the beginning, but instead of the gateway, build forge,
2. With the probe that built pylon you should seek your enemy's base and build a pylon in it,
3. If you haven't found it before the forge starts warping in, use another probe to scout the map,
4. After the pylon has been completed, build 2 cannons behind it (if you're playing zerg or protoss) or in front of it (if you're playing terran).

You can also build a few cannons at your base for defence. After the first 2 attacking cannons warp in, keep advancing in your opponent's base until you win.

Most od the people cannon up, also, most of them do it wrong by cannoning up just the ridge or a bridge to their base. In this case a drop or an air rush can easily kill them off. My basics of cannoning up is to build a packed base with 4-6 cannons that can be reached only by one unit at the time. Building a cannon in worker line can be very useful (just one, or maybe two). The other important thing of cannoning is to keep all of them close, but still enough to protect the whole base. Usually the best way is to build 4 cannons that would be surrounded with nexus, gateway, forge and a few pylons, this way, your enemy will be able to reach you, but he'll be dead by then. Cannoning up is not recommended if you go for early expansion.