WPEngine is about the gold standard here, and they have a very good migration plugin in the wordpress store. "Tight budget" doesnt really define a dollar amount. If you can afford them, use them. If you are talking under 20/mo, I used hawkhost before when I didnt have the time to move accounts around. They are about the only one I can think of in the "cheap, but good customer service" category. Im not sure if they still migrate accounts, if you are on cpanel, then prolly.