I am looking at getting a trademark through Legalzoom for a new product my startup is about to launch. If the logo for the product includes the name of the brand and the product, do I really need to also trademark the names on their own?
You can apply for registration of a trademark used in commerce in the United States (that is, used to achieve actual sales across state lines) through the United States' Patent and Trademark Office at www.uspto.gov. You will find a number of useful references on that site that will help walk you through trademark basics. While getting advice and counsel from a knowledgeable attorney would be a good idea, you may not need much help filling out the (fairly straightforward) trademark registration forms. For this answer, I presume that by "getting a trademark" you mean "registering a mark used in commerce for protection".
More substantively: a trademark registration protects the use of the mark itself in commerce. Simply put, if you register a mark in the logo, you are only protecting the logo. As a practical matter, this may well be enough - the particular facts of your circumstance will control. But the bottom line is you will have much more reliable protection over the brandname and product name if you separately protect those, particularly if they are ever used as word marks separate from the logo.
One way to think of this is to consider the "Circle R" symbol: ®. That symbol may only be used following registered trademarks. So consider if you register the logo but not the words, and had an ad like this:
LOGO® - We make the best stuff! If you're looking for the best, buy GO.
("LOGO" is your logo, here, and "GO" is your brand name.) You would not be able to use the ® after the "GO," which implies that it is open season on the use of that mark.
You get pretty significant protections on a mark whether or not you register it at the PTO. But if you have uses in commerce, you are typically better off getting as many separate registrations as possible.