How do I trademark and copyright my logo and wordmark?

Henry Thoreau once said: "It's not what you looksure at that matters, it's what you see." So let's clarify some terms and definitions first, to make  we are on the same page here and that you see the same things I see.

Trademark in Vietnam

When speaking about "logos" and "word marks", it is important to realize that those terms are not interchangeable, from a trademark attorney's perspective. There is a clear legal distinction. By definition, a “word mark” includes words only. If a mark includes some graphical/design elements instead of, or in addition to, words, then a trademark attorney would refer to such a mark as a “design mark” or a “logo”.

So if you have two separate trademarks (one of which is a word mark and another is a logo/design) then those would be two separate trademark applications. That applies even if the logo incorporates the same wording that you use separately as a word mark. Assuming both the logo and the word mark are used for exactly the same goods/services, then, yes, you would claim the same International Class(es) in both applications.

If a design mark / logo crosses a certain originality threshold, then it can also be eligible for copyright protection. The threshold is not overly high, but it does exist. If you merely apply some minor stylization to your word mark (e.g., the words appear in an unusual font) or your mark consists of a simple geometrical shape (e.g. a circle) with some words inside, then there might not be enough originality there to claim copyright protection. On the other hand, if a logo incorporates a drawing of animals, objects, people, etc., then it will be much more likely that the originality threshold has been crossed, and copyright protection will automatically attach to the logo as of the date it was created. There is a maze of rules as to which logos are copyrightable and which are not, but I think you got the overall picture.

As to word marks, generally they cannot be copyrighted, but like with everything in life, there are exceptions. But that is a separate long story.

A copyright owner is not required to obtain a copyright registration for his/her work (since copyright protection is automatic), but it is a very good idea to get a formal copyright registration as that provides significant benefits (i.e., public record of ownership, which can negate certain defenses an infringer might have, access to federal courts, a certain level of protection against importation of counterfeit products, etc.)

Related post: 

What is the difference between a trademark and a wordmark?