I would suggest the following
1) Get a tourist visa to a western country, anywhere in Europe. Perhaps pick a place where you can speak the language.
2) Get on a plane. ideally from Dubai.You can go there without problems, and there are more flight connections there. Also, you are not likely to be subjected to quite so rigorous checks when leaving from there. DON'T take tons of luggage. It's only stuff, and stuff can be replaced.
3) Once arrived claim assylm on the grounds that you may be physically harmed for renouncing Islam.(A week or so after arriving may be a good idea) It is best done by seeing a solicitor who specialises in Immigration.
If you have money in Saudi Arabia, go to an international bank in Saudi arabia, like HSBC, it's a UK bank, but with big shareowners in China and Hongkong. They have branches in SA. Open an account with them, and put your money there. SwissPrivateBank.com is also a good option for that, but you need to have a fair amount of money to work with them. Don't take a large amount of cash, you may get hassle with costoms.
Once you have arrived in Europe you can ask the bank in Saudi Arabia to transfer your funds to a European bank account. You need to to that quickly,and ideally before you declare your intention to claim assylum.
When calling friends and family in SA be sure you uses Skype and a proxi server, because you don't want the friendly neighbourhood Saudi religious police get on a plane and pay you a visit.
Settle in a big city, one with other Arabs living there. Consider changing your name.
You can also go to a number of countries without a visa or with visa on arrival. New Zealand? Equuador? South Korea? (Now, that is a really nice place to live) Thailand? There are direct flights from Dubai, and nobody in SA needs to know you are going there. If then you want to go to Europe, you can get your visa there, and take a direct flight out.
Here is a list: