I was going to write my answer to Comandante, but marx above here pretty much nailed all I was going to add to the thread. Point for point, just clear and sensible informations.
350 GBP will put astrophotography out of the equation pretty much but they should allow for a quite decent piece of kit if you only go visual, unless as marx says you don't already own at least a DSLR (not sure how the more recent mirrorless designs fare when attached to a scope, I don't have first hand informations on that). There are also "cheap" dedicated planetary cameras (and for cheap I mean in the 100-200 £/€ range, but they aren't much more than glorified webcams below that price range), but a DSLR would be a good all-around tool and also probably easier to use.
They'll say you that when choosing a scope, aperture is paramount, and it is indeed in some ways, but there are other factors to take into account. So adding on what marx already wrote:
- Newtons are the best bang for buck, but they require some moderate skill to be used and mantained. It's an open reflector design (light enters the tube and gets reflected by the large mirror on the bottom on the small mirror at the front, that will converge it to focus on the eyepiece), meaning that mirrors need to be kept clean and collimated from time to time (not a daunting task, but still something that takes time and knowledge). They also tend to get quite bulky once you get past the 5"-6" of aperture.
Something to take note of, the secondary mirror in front of the light path means that part of the entering light gets obstructed by it, leading to a slight loss in light gathering capability compared to a refractor of the same aperture.
- Refractors are the oldest type of telescope (same as Galileo's one), great all-rounders, virtually zero maintenance aside from keeping the front lens clean. Great for photography but with the caveats cited by marx, achromats will have optic aberrations, apochromats will give exceptional images but you won't see them, 'cause you'll have sold both corneas to afford one probably.
They are limited in available apertures to 6" (larger than that and they get quite tricky to produce for the mass market), but having no obstructions on the light path they compare very well with Newtons and Catadioptrics (Maksutov/Schmidt-Cassegrain and the likes).
If you go the refractor way you'll have to choose carefully your visual priorities: short tubes are great for wide fields of view and deep sky photography but they'll lack in magnifying power unless going quite heavy on eyepieces/barlow lenses, and they'll add lots of aberrations at high powers. Long tubes on the other hand are great for planetary and high power work (like splitting binary stars), but with a narrower field of view and less suited for deep sky photography.
- Catadioptrics like the Celestron linked above (a Schmidt-Cassegrain) are a bit of both worlds, they are primarily a reflector design as the Newton but they also use a front lens to cancel out chromatic and focus aberrations. Due to the way they fold the light inside the tube they can be made a lot shorter than the focal length would suggest and this makes them just about the best for portability. The view quality is similar to that of apochromatic reflectors at a fraction of the price, the downside being that having a frontal obstruction images will be dimmer, and they also have a very narrow field of view that makes them optimal for planetary viewing, but not the best for low power, wide field views of starfields and large deep sky objects. They usually cost halfway between Newton of similar aperture and high-quality refractors.
And this pretty much covers the main things to know about different designs. But that's just half the instrument! Never underestimate the mount you're going to put that on, any scope is usually just as worth as the mount it gets used on. To put it shortly, Alt-Azimuth mounts are the easier to use (horizontal movement parallel to the horizon) but are not suited for long exposure photography because the object will rotate in the field of view while you track it. Equatorial mounts are more complicated to set up and use (horizontal movement needs to be parallel to the equator, and they need to get perfectly aligned to the North Pole for this to work) but they are the only ones allowing long exposure photograpy. The more sturdy the mount the better of course, some scopes comes with decent mounts out of the box already, some not, it's a thing that need to be checked before buying.