You can’t patent it, but you can trademark it! Most will tell you it's smart to trademark at least your name and logo in order to prevent others from ripping off your startup and be provided a legal recourse if this ever happens.
Although you can claim ownership of your trademark without formally registering with the United States Patent and Trademark Office (USPTO), your rights are limited to the geographic regions where your trademark is used. Also, if another company infringes on your name then you will likely end up spending more $$ litigating in court by failing to register. Thus, formally filing for trademark protection will afford you maximum legal protection from the start.
You should consider hiring an attorney to perform your trademark application. Not only will an attorney assist in navigating the process, but s/he will also provide you with much more. For example, they will respond to correspondences/rejections from the USPTO, classify your goods/services, or help you understand the scope of your intellectual property rights.