Some thoughts (not NH specific):
(1) "Protecting your name of your future business" really is a matter of deciding what name you are going to be using as your business and start using it. Most states allow you use a "d/b/a" (doing business as) designation, so there is no need to incorporate to start using your business name. Many states don't require you to even register the d/b/a but you'd want to check and see what your state requires. The only financial investment that you should consider making at this stage is to obtain the domain name of your business. You can get that pretty cheaply through godaddy or another company--and be sure to look around for discount codes online. Then make a free blog using blogger or wordpress and forward your purchased domain name to the location of your blog. Your domain name provider will have a "help" post on this but it is really quite simple. The end result is when anyone types "mycustompottery.com" into their browser, they will be taken to "mycustompottery.blogspot.com" automatically and very few people will even notice.
Another reason you want to start using your business name asap is that you want to establish a sufficient prior use of the name so that no one else can claim the trademark. Anyone can create a company of the same name so that is of no real legal protection. What you want to do (and what the big companies do automatically) is to show that you are the first to be using that name. Then, even if someone else registers the trademark, you have a claim of a superior right due to your first use.
Finally, you want to use your business name as much as possible on the web so that "all roads lead to you". If a customer has one of your pieces and types your business name into google, you want to have your site pop up on at least the first results page. No one goes to page 2 of google results so you need to get on page one. That is done primarily by using your business name link as much as possible, having it as the address when you post comments on other sites, having it in your post footer as Mark C and others do in this forum, etc. The more use, the better.
(2) Getting involved in the "state" - your main concern should be finding out when you need to declare your sales on your state & federal taxes; and when and if you need to collect sales tax. Your local SBA office will be able to help you with both of these or get to an accountant. Heck, do both! The first is free. The second may charge you for a consult or may not, but a good accountant will help you balance your pottery related income from your pottery related expenses, etc. when possible. The main purpose of an LLC is to provide a limitation of liability (hence the name, Limited Liability Corporation) so that if something bad happens (or you run up a bunch of debts), the responsible entity is the LLC and you and your home and personal finances are protected from lawsuits. There is rarely a tax advantage to having an LLC as the vast majority of LLCs are taxed as "S" corporations, which means you (as the corporate shareholder) and not the corporation will be paying the taxes on your earned income. LLCs are not difficult to set up, but there are some filing costs and you have to do annual reports, might have a separate income tax report to file, etc. Fun paperwork that all potters love! ;-)