Brand names, logos, and slogans can all be trademarked. Prioritize registering your brand name first since it holds the most value. Logos deserve separate protection if they're well-established, but skip the slogan initially to save money.
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When registering a trademark, there are 3 main things that you're going to want to consider trademarking. The first one is the brand name, the second one would be the brand's logo, and the third one would be the brand's slogan. Registering all 3 would be a little costly, so if you're trying to be a little bit more cost-effective, then at the very least you should register the brand name. In many situations, the brand name is what has more value tied to it, whereas with the logo, usually business owners change that pretty frequently. So at the very least, you're going to want to register the brand name. If you have a very established logo, then definitely you're going to want to also register that logo separately.