How exactly do we register a trademark?

I would suggest you start with the tutorials on the website of the U.S. Patent and Trademark Office at http://www.uspto.gov/trademarks-getting-started/trademark-basics and then you can decide if you want to prepare and file one or more applications on your own or if you want to consult with a trademark attorney. Working with a trademark or IP attorney is strongly recommended for a start-up for many reasons, including reasons beyond merely registering your mark or marks, and I am not just saying that because I am a trademark attorney.
The cost is a minimum $225 per class of goods/service filing fee to the USPTO and the fastest the USPTO will be to register a mark is probably around 7 to 9 months but that assumes you file everything correctly (which a trademark attorney would do) and that you are using the mark because a U.S. applicant cannot register a mark until it is using it in U.S. Commerce. The meaning of "use" is a legal term of art and understanding this legal concept is another reason to work with a trademark attorney. Probably the single most important reason to consult with a trademark attorney is to conduct a clearance search on the mark to ensure it's availability and assess the risk of proceeding with that mark. Working with a trademark attorney might make the process cost around a few thousand dollars, depending upon what you want done and how many marks are involved, but could be well worth it considering the costs you might incur if you adopt a mark that is not available; you make an error in the application that is not correctable; or you obtain a registration that does not properly cover your mark or your goods as the mark is actually used in U.S. Commerce.