Should we register the trademark or copyright for our app?

Here are some definitions of patents, trademarks and copyrights to help you out:
Patents – a patent is a right established in the U.S. Constitution and granted by the U.S. Patent and Trademark Office (“USPTO”) toan inventor “to exclude others from making, using, offering for sale, or selling a particular invention throughout the United States, or importing the invention into the U.S.,” for a limited time in exchange for public disclosure of the invention when the patent is granted. The limited time is usually about 17 to 20 years. The typical patent is a “utility” patent which covers any “useful process, machine, article of manufacture, or composition of matter, or any new and useful improvement thereof.” However, a “design patent” can be procured from the USPTO for any new, original, and ornamental design, usually of a particular product. A patent requires that the invention is “novel” (never before been achieved) and not “obvious” from any existing invention or known technology. Thus, a patent is extremely powerful because it provides an exclusive right to offer an entire product – i.e. a monopoly - but only for a limited amount of time and only if truly new to the market.
Trademark (Service Mark) - A trademark is “a word, phrase, symbol, or design, or a combination thereof,” that identifies and distinguishes the source of the goods of one party from those of another. A service mark is the same as a trademark but identifies and distinguishes the source of a service instead of goods. The term “trademark” is typically used to collectively refer to both trademarks and service marks (and will be used as such throughout this blog). A trademark must therefore be associated with a particular product or service offered in commerce and only prevents the use of that name, symbol, phrase or design from being used with a later seller’s similar product or service in order to prevent consumers from being confused about who originally offers that product/service. A trademark right does not necessarily stop the sale of another product outright. One establishes a right to a trademark as soon as the mark is used on a product or service in commerce, but registering the mark with the USPTO or a particular state government helps to protect that trademark and enforce it against infringement.
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Copyright – A Copyright is a form of protection also established in the U.S. Constitution and federal laws for the authors of “original works of authorship” “fixed in a tangible medium of expression” such as literary, dramatic, musical, artistic and other intellectual works. A Copyright prevents another from directly copying one’s original work if recorded in some tangible form, such as a writing or drawing, an audio or video recording, an architectural rendering or a live performance and it does not matter if the tangible form is handwritten or typed, published or unpublished, electronic or on paper. However, copyright infringement requires that the copy be virtually an exact copy, not merely the idea or concept expressed in the work. The Copyright right is established immediately upon creation of the work, if the work is sufficiently original, and does not need any registration or grant by the government. However, registration with the U.S. Copyright Office, a part of the Library of Congress, is necessary in order to sue another for infringement and also to collect a greater amount of damages set forth in the law, known as “statutory damages.”

In your case, your could register the name and any associated logos that identify your app to broaden your trademark rights but you get rights in the mark as soon as the app is on the market. You could register the copyright in any novel designs, text, programming code and artwork for the app. The patent can only cover technological aspects of the software that represent an advancement over existing technology and tend to be rather difficult to obtain for apps unless the app has an advanced function.
You really need to consult with an IP attorney to get a more detailed answer for your specific situation.

My suggestion is to file a trademark (word or logo) of the app before launching it into the market