What is the process for registering a logo together with the word trademark?

If you want to register both the logo and the name of your brand as a trademark, you can go about it in two ways:

1. You can register the logo as a so-called figurative mark and the name as a wordmark. This option offers a higher level of legal protection; however, bear in mind these trademarks would have to be submitted as two separate applications, meaning paying the fees twice. If your applications get approved, you will own two independent trademarks.
2. If your logo includes your brand's name, you can choose to submit only the logo. In that case, the words contained within the mark will receive some level of protection, although not the same as if it were a wordmark (where wordmarks protect the name itself, figurative trademarks protect more the visual stylisation of the mark).

After you make your choice, the process is essentially the same - you will submit your trademark application(s) to the intellectual property office, which will assess it. If you submit one for the logo and one for the word trademark, they will be evaluated independently of each other.