I would like to trademark a mobile app/website name. The name and slogan would be on both the website and mobile screen where the app is downloaded. Can I register the slogan along with the name? For example, "Name, Slogan". If yes, would I need to file the mobile app as a TM or SM?
A registered trademark contains exactly one trademark. If you have a name, a slogan, and a name + slogan; then you have potentially three different trademarks that could be registered. If you want to choose just one then, then you should consider factors like: which trademark will my customers use to find me, which is the most suggestive, what advantages or disadvantages might exist for enforcement/defense purposes.
Your trademark can be anything you would like, so long as it sufficiently suggestive or has acquired distinctiveness. So, if you register both the name and the slogan, then that entire set of words is your trademark. Each one, by itself, would not be a registered trademark. So, after filing for the words + slogan trademark, you would strive to always use them together.
Some more factors to consider when selecting your mark would include: how would customers spread the word about your product? Would they use the name, slogan or both? How will customers find you? How are you marking your product. Is your name or slogan alone sufficiently suggestive? Has any part acquired distinctiveness through use?
In general, a small business should strongly consider first obtaining one trademark, words only, around the set of words that customers would use to find the small business's product or service. This may not be the easiest trademark to register, but, it would likely be the most valuable or helpful if enforcement or defense of the trademark becomes necessary.
Whether a mark is considered a "trademark" or "servicemark" depends completely on wether goods or services are offered. Goods have trademarks. Services have servicemarks. A mobile app might be classified in class 9 as downloadable computer software. However, it may also be possible that some mobile apps that derive their functionality primarily from a network would be considered software-as-service. It depends on how you conduct your sales.