climbing shoes
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.
Climbing shoes can be comfortable! But, that is in comparison with other climbing shoes, and not with everyday shoes. I believe that climbing shoes will never be as comfortable as everyday shoes.
If you manage to find the right pair of shoes for your feet, though, they can be very comfortable, considering they are still climbing shoes.
For me, the best way to break them in faster and to make them less painful was to just wear them. The more I wore them, the softer they became. One thing that really helps is to heat them up a little bit before putting them on with a small heater or fan, just enough to soften the rubber a little bit, not too much though to not damage the shoes.
Another thing that really helps and that is very simple to do is to just bend, twist, and fold the shoes for a minute or two. This really softens up the rubber and makes it much easier to put on and less painful to wear.
It is very common for them to hurt, but, they are not "meant" to. They are meant to be very tight with no dead space, and this causes them to be painful. They don't always have to be painful though, it depends what you are aiming for with the shoes, what kind of climbing, what kind of routes, etc.