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I would suggest you stick with any area near BTS or MRT lines because it will give you easy access to other parts of the city.  Some area's are better than others because they offer close by supermarkets and western food restaurants within walking distance.

Your budget might make it necessary to find a place a bit further away from BTS and MRT because most places will charge a bit more if not leasing for 1 year. You can check in hopes of finding a good deal.

I for one like the On Nut area because it has many new condo and apartment building all within walking distance of BTS and two supermarkets.  You can check out which is where I live.  The thing that makes this place stand out is on premise pub that offers reasonable priced food and drink.  Its become a meeting place for the tenants and unlike other places I've lived I actually know many of the other tenants because of the pub.

Ok, I think you're right. Maybe I should consider raising my budget for this.

The 'problem' is, I have a feeling one can rent good flat on good location for about 12-13k Baht, which fits into my budget if one know what he's doing. Generally, I would like to avoid any middleman type agencies that add up to the final price.

If you came to my country and tried to rent through some agencies or similar, they would add up about 25% on the real price. This is alright because you have less hassle, especially for someone like me who lived in cities where population don't exceed 300k, and moving to bkk with 12-13 million people. I understand stuff works differently there, but still.. I would like to avoid it if it's possible. So any tips or information or whatever regarding this is also welcome.

Thanks for the links. I also found some similar sites, with good deals (at least it looks like that) - easyhomes.tv.

As you say the best deals are dealing directly with owner of property.  One thing you should concern yourself with is return of deposit when you depart.  Some people never see their deposit again because owner finds some excuse not to return.  I have never experienced that problem renting where the entire building is owned by one person.  If you keep the place clean and vacate without damage you should have no problem getting deposit back.

One need to be careful with the owners and everything must be clearly stated like when they return the deposit money.  Some may take weeks or months to return the deposit money.  Make sure you read the contract carefully and if there are damage in the apartment, have photos taken and have it stated in the contract.