Create game LoGos for my site

Hey there! That's a complex topic, but I'll try to give you some direction. Adobe Photoshop can be used to create digital illustrations and artworks, but Adobe Illustrator is better suited for logo design. It is used to create vector based graphics, which are more appropriate for print and use accross many different devices and mediums. The illustration you linked isn't exactly a "logo" in the traditional sense, as logos are usually lower detail and cleaner. You are going to get closer to that type of graphic if you search for "vector illustration".

You can start quickly by heading to YouTube and searching for the following: "adobe illustrator beginner course". These days there are A LOT of resources and people making courses and tutorials available for free. Once you get the hang of the basics, you can experiment more freely and try to create something on your own.

Graphic design and illustration are huge fields with many possibilities for specialization. People study and practice for years until they can create high quality products consistently. So manage your expectations, don't think you will master it after one course, but keep at it and you'll get better!