It is the most common belief to get an architect before the contractor. However, it can be beneficial either way for a client. Both ways have pros and cons and it is generally up to the client. For instance a pro benefit for choosing an architect first. They will have quite a bit of knowledge on the inner workings of what the city or county might and might not allow. Where as the contractor only has limited knowledge of the process, because the plan reviewers deal mainly with architects.
One pro benefit for choosing a general contractor first. For instance, they can give you budget numbers closer to what you are needing and on par with your needed budget range and work with you throughout the project. This is very beneficial for first time building clients. Also, the lenders rely on budget numbers and the clients building scope is a serious reflection on the clients finances. Typically contractors do not charge for some additional guidance on what tends to be the most cost effective solution with a project.
There are many pro’s and con’s for either side. However, if you are dealing with a budget and need to stay near to it. A contractor would probably be the best choice. That is to say, you will still need an architect most likely, but the contractor should be able to help discuss ideas to keep the project in the budget.