Here is the most simple explanation I can give:
1. You must have the luxury: it must be improved. Use a worker and improve the luxury tile. Note: when you settle a city on a luxury, you will automatically get the resource if/when you have the technology required to improve the resource.
2. In the diplomacy drop down, click "diplomacy overview." See which AI has money. This will also show which luxuries they have: you can not trade a luxury to an AI they already have.
3. When you have decided on a trading partner, choose that leader and choose trade. If you have a luxury they don't, on your side you "Luxury Resources" will be illuminated, click it and select the resource you want to trade.
4. Either make an offer by selecting the other AI's items, or ask what the AI will give you for the resource. If the AI has 240 gold, this is generally the top price you will get for a luxury. You may have to settle for gold + GPT if the AI does not have enough gold.
Notes:
-Even if the AI says there is no offer they can give which is fair, you can get them to give you less, which they will appreciate, and can be more helpful than getting full price 10 turns later.
-If you have only 1 copy of a luxury, you will lose the +4
from the resource when you trade it (unless you are the Netherlands). Try to trade luxury resources you have more than one of. Also, time your trades of unique resources (for example: trading gold immediately when mined to buy a settler and getting the additional +4
in 30 turns when the city has grown and it is needed.