If you're only budgeting a few thousands dollars for your website, don't expect to have a website with an award winning design or extreme amounts of functionality ranging from user access control, e-commerce, and custom APIs. It requires many years of experience to be a quality website builder or effective programmer, and they deserve to get paid for their time just like you do. Many web companies will still work with smaller budgets, but the expectation on what's delivered should be lowered as well.
If your budget is low due to actual lack of funds, you would probably be best served by using one of the services such as Wix or SquareSpace to get something that looks good that you're able to manage yourself.
Building a website or web application is a creative process as much as it's an analytical process. It takes time to get everything working the way a user would expect it to work. There needs to be ample time for both the client and builder to communicate back and forth and even not work on the project from time to time. This will allow each party to think of new ways for the project to look or work.
I can't count the number of times I've had to work on 'emergency' projects that weren't thought of until a week or even days before a big trade show or release of a new product/service. It can be very stressful to work under those conditions, and you're guaranteed to not get the best product in those scenarios.
'But overseas developers are so much cheaper!', you say. And that is correct, it is cheaper to work with someone in Pakistan than it is to work with someone in Nashville, Atlanta, or any other major city in the US. There are countless reasons to work with someone in your own country, but the most prominent factors that will effect you directly are quality of work, ease of communication, and reliability both today and in the future.
Most major software companies have a team of developers overseas, and that works for them because they have the infrastructure in place to find, manage, and retain talent on the other side of the planet. You probably do not! So trust me, the adage is true: 'you get what you pay for'. And if things go south, it's much easier to find Harry Smith at 202 5th Ave than it is to find this guy:
Mr. Nasratullah Khan
House No 17-B
Street No 30
So there you have it. A few quick tips to think about, and I'll have more to come in the future.