I would argue that your business plan should come first. Your business plan requires a large time investment and will be the foundation of your launch and the direction of your new business of the first year or two.
Trademarking or registering the name of your business is secondary. Who knows, you may discover a better name for your business as you explore your concept at a deeper level during the creation of your business plan.
If the name of your business is the best thing about it, you are in trouble. A catchy name if cool...but it means nothing if you do not deliver goods or services that people actually want.
I hope this helps!