How can I make my climbing shoes more comfortable?
First of all, get the right shoes in the right size. The most common mistake I hear people make in my gym is that their shoes are way too small, and they don't even know it. In that case, their shoes will probably never be comfortable and may always be slightly painful.
If you get the right shoes that match the shape of your feet, and in the right size, you could actually find some very comfortable climbing shoes. But, you have to keep in mind that these are still climbing shoes, and that they were not designed to be comfortable, they were designed to give you support for climbing, for performance.
If your shoes are too small, it will be difficult to make them comfortable. If they are the right size, however, and they are the right fit for you, then you can try and break them in a little bit more if needed, there are many methods for this.
Something that I like to do is to slightly twist and bend them before putting them on. I really notice the difference before doing this and after. It really just takes a minute or two to just bend them and twist them slightly. Not too much to the point where you damage them, but just slightly to loosen up the stiff rubber. After this, the shoes usually become much more comfortable, and after climbing in them, they sort of open up and are just fine.