You are not required to register your business or organization name as a trademark, but doing so offers two major advantages: (1) it puts others on notice that you are using the name, so new businesses do not adopt it by accident; and (2) in the event of a lawsuit, it makes your case much stronger because your trademark rights are clearly established once you register. You do not need to register the mark before you start using it. While the federal system gives that option through "intent-to-use" applications, your state may not, and if you are ready to start using the name there is no advantage to waiting for registration.
Once you have decided to register your business or organization name, and preferably actually taken at least some concrete steps toward using it, you should file the necessary forms to register it in your state and/or with the US Patent and Trademark Office (USPTO). Registration is not free (federal registration costs at least $275, and as an example, registration in Massachusetts starts at $50) and may be subject to certain limitations or requirements, such as actually using the trademark in connection with goods or services. Federal registration is the most powerful, since it presumptively applies throughout the country and is most likely to be noticed, but a state registration still provides greater rights than if you do not register your mark at all and makes it likely that a new business will discover your use of the name.
State procedures vary. You should contact your state authority responsible for trademark matters for details.
You can apply for a federal trademark registration online. The USPTO estimates that filling out the form should take about 15-20 minutes. A federal application must contain at least four things, in addition to the filing fee: your name, your address, a clear drawing of the mark (which can be automatically generated from text if you do not have a logo), and at least one category of goods or services in which the mark is or will be used. Alternatively, you can file a much more detailed application and receive a discount on the filing fee. You can choose more than one product, service, or industry for your trademark, but you must pay the filing fee for each one. The USPTO has predefined categories of products and services that they strongly suggest you use, but you can also type in something else if your product or service does not fit in its categories. You may want to consider filing for more than one category. If your business is not clearly confined to one area, doing so would give you flexibility to expand into new products or services in the future. On the other hand, aside from the added filing fees, the downside to doing so is that you are more likely to conflict with someone else who is using the mark or a similar one in another field or industry. Furthermore, if you do expand in the future, you can apply for a new registration (unless someone else has already registered the mark in that area in the meantime).
After you file the registration forms with the USPTO, within about 3-6 months a USPTO attorney will examine and research your application. There are several categories of marks that the USPTO will refuse to register, including "immoral, deceptive, or scandalous" marks (such as including foul language), those that disparage or falsely imply a connection to other people (the Washington Redskins are in danger of losing their trademark registration over this), and marks that are confusingly similar to others that are already registered. The USPTO attorney may contact you to resolve any issues in your application. If the USPTO approves your application, it will publish your trademark in its Official Gazette, and anyone who is already using the mark may oppose the registration. If no one opposes within thirty days, the USPTO will register your mark. All told, the process can easily take 1-2 years, but once it is approved your rights date back to the day you filed your application. You will need to renew the registration every 6-10 years to indicate that you are still using the mark.
If this process sounds very involved to you, you can also hire an attorney to do it. Trademark registration is usually fairly straightforward for attorneys who specialize in it, so it is relatively inexpensive.