> a photon is a particle, that travels in a wave
No. This is just wrong. Completely. You need to make your brain unlearn this.
Here is a toy model you can use on your journey... Think of the photon as a cheshire cat. You cannot see the cat, if you try to perceive it completely, it will vanish. You can, however, ask it questions. If you ask it "what is your gizifa", it will say "10". Or in this case, you can ask "what is your momentum", and it might say "5".
Asking certain questions will upset the cat and cause it to change all the other values just to piss you off. So if you ask it what its momentum is, the answer you *might* have got for its gizifa will now change. These values also change on their own over time. So even if you know that its location is 2,7 now, when you ask it again later you will get a different answer. No, this is not *because you asked* (another common misconception), this is inherent to the way the cat works. Some of these values are conserved (like electric charge), others are not (like location) and others are linked together (like momentum and location).
Photons are not particles. There are no particles. "Particles" is the term we use when we refer to these things when you keep asking them what their location is. If you do that, they will give you nice answers like 2,7 and then 3,7 and then 4.7, and you'll go "oh, this thing is travelling along positive X, and it's a point, so it must be a particle!". But the problem is that if you ask it different questions, like its position and the location, then any semblance of particle-like behaviour will vanish. You were fooling yourself, ITS NOT A PARTICLE. Neither is an electron or a proton, or anything else. They're just quanta. It's all quanta.
> It has some pressure when it shines on an object
This is also incorrect.
Newton thought momentum had something to do with mass because he only had large objects to work with. Shotputs have a lot of momentum, and so do planets. But in the "real world" of quantum, momentum is just a number. It's a number like any other, like energy. It's not related to mass. You ask a quanta a question and it will give you an answer. If you ask a photon its mass it will say zero. And if you ask it its momentum it will say 5. These questions are orthogonal, they don't have anything to do with each other.
So why does it LOOK like momentum has something to do with mass? Because the momentum of any one quanta is tiny, so in order to be measurable at macro scales, you need a WHOLE LOT OF QUANTA. It's very easy to make a ball of protons and electrons, because they attract each other. So you put a bunch together and call it a shotput and notice that it has a lot of momentum. But the fact that it has a lot of momentum isn't because it has a lot of mass - it has a lot of mass AND momentum because it *has a lot of quanta*.
It is much harder to make a big ball of photons, they don't attract each other. If you did such a thing, you'd find it had just as much momentum as a ball of matter, but still has no mass.
> and given off an infinite amount of mass/energy
No. Energy is also a measurement of the same sort, it's just a number you can ask for. It has nothing to do with "speed". Some of these values you ask for are more interesting than others because they are concerned, but that's not due to quanta, that's due to the shape of the universe.