You might come across a fun looking player who is talking about betting and payouts with BIG cashstacks. These aren't players, but automated bots controlling an account. They scam players by taking large stacks of cash used as wagers for their fixed automated gamble. Stay away from trading with these accounts. They will rob you for all your account is worth and then make you come back for seconds.

Players will claim that they will double the amount of money you put into a trade. They trick you by coercing you to complete a trade with them and then will log out straight after you give them your coins. Sometimes they will double a small amount of money to get you to trade a lot the next time. Watch out for these scammers.
Begging scammers will pray on the generous by pretending to be a player in need and then will continue asking for money straight after.

Players will claim that they can trim your armor for you. This is a lie made up to steal your valuable armor. Trimmed armor comes as a reward from completing clue scrolls. There is no way for players to craft it.

When trading items, it's a good idea to avoid direct trades with players. There is a counter in the trade interface that estimates how much items are worth, but don't be fooled! Some items will have a high in-game price, but aren't really worth much of anything since the item isn't in demand. Players will tell you they have an item worth 50k, and sell it to you for 30k. Then you will be stuck with an item you can't sell because nobody wants. Stick with the Grand Exchange for fair trades. Its location is highlighted in green on the map.

The wilderness is a dangerous place! When you go there, other players can kill you depending on how far their combat level is from yours. Normally, when they kill you, they can take all of your items except for your three most valuable items. If you attack another player first, you will get "Skulled" meaning you will lose all of your items on death. I repeat, you will lose every single item in your inventory! You can go into "Settings > Controls > Player 'Attack' options:" and set it to "Always right-click" in order to prevent your character from attacking other players accidentally.
When crossing into the wilderness, you will be given a warning. This will also happen if you try to teleport into the wilderness, so make sure you read the pop up window carefully!

All worlds labeled in red are PVP enabled worlds. If a player asks you to meet them in a pvp world, be wary. They may try to kill you for your items or even trick you into getting skulled!

If any players have any suggestions for sections to add to the guide, please leave them in the comments! I can include your tip along with credit given in the section. Thanks, and happy scaping!