I'm a web designer. I build websites on WordPress for small and startup companies. I take care of everything; graphics, content, hosting and domains. Logo design is all part of the service.

Brush tool

Anyone who doesn't understand branding and sets up a website may struggle to be recognised on the internet. It's important to have a distinctive mark that sets you apart. A decent logo is the key to this.
The logo should be a vector image made at 300dpi in a .png or .bmp file to avoid the distortions you get in .JPG of .GIF files. If you've no idea what that means, contact a designer.

While graphics tools give you the means to make your own images, knowing how to make those images work as logos isn't something you can pick up overnight. You have to understand how logos work.

Logos have to be clear and distinctive at different resolutions. At different sizes, flaws in design are revealed.
I've seen some websites where the owner didn't bother with a logo of any kind, not even a bit of text to let us know what the website is called or who owns it.

That's taking the content-first approach too far.
Confronted with stupidity
The purpose branding is to mark the item as belonging to the owner alone. The principle is that the particular mark or pattern is identified with you. The image or symbol and the font we choose are to represent your company as BEING a particular thing.

Branding
In this example, "Barry the Bricklayer" is represented as much by the font we've chosen as by the logo image.

Barry the Bricklayer, an example logo

This is important to note because failing to adopt and stick with a brand can harm your business by making it hard for you to stand out from the crowd.  

You may have seen this in action when a major company makes changes to its logo, often at vast expense, and people flip out and demand that the old one is restored. Why? Because they've got to change the way they recognise the brand .
Me being serious

The text should be proportional to the image and the image should be clearly visible at different sizes.

The trick is not to make it too detailed. Details don't show up at tiny sizes.

"But I can just have plain text if I want it?

"I don't need anything fancy, just a simple website that lets people know who I am, what I do, and where to find me."
Mr. Know it all insists

"I'm not planning on world domination or anything. I'm a small businessman and honestly, I don't like too much fuss.

"If I don't need anything fancy, I won't have it.

"And honestly, I don't want to spend the money.

"Do I have to trademark my logo or anthing like that?"

Nope, it's not generally necessary.
Me, smiling
Trademarking a logo gives you alone the right to use it, but prior art applies so only big brands tend to bother with it.