A 5.1 system means there are five satellite speakers and a sub. The '5'-bit is a centre speaker then front L&R and surround L&R speakers. The '.1'-bit refers to the subwoofer. The centre speaker is the most important of the 5 satellite speakers. This is where all the dialogue comes from.
Speakers positioned in the corners of the room sound terrible. I know it might look neat, but trust me, you want the front L&R speakers at least 1ft (30cm) away from the corner edges of the room. They need to be about ear height when you're sat down. The surround speakers shouldn't go in the corners either; and although some call them 'rear speakers' they aren't really that either. They're side surround speakers and so should go on the side walls about inline with where you're sitting. They can go higher than the fronts. Put them at ear height when your standing.
A £300 budget won't get you a separate AV receiver + speaker package. But £350 will.
At £300 or less you'll be looking at either a soundbar (with or without sub), or a Home Theatre in a Box (HTiB) system where the main unit has a Blu-ray player built in and there's about a gazillion features. They're a poor choice. The Sony BDVE2100 (£220) is a good example of this.
This system offers Bluetooth and 3D and Smart features and NFC and what appears to be masses of power (800W) and Wi-Fi and Dolby True HD and DTS Master Audio and ... and... and.... But the problems are:
1) there's no way to connect the HDMI-out of your consoles to it directly so all this processing is limited to the internal disc player
2) the internal BD player is often the first thing to die, and when it does the cost of repairing it is way out of proportion to the price of the unit, so its effectively dead
3) nothing in the system is upgradeable. You're stuck with these speakers and/or the sub and/or the head unit
4) the power output is a work of complete fiction. The unit draws a maximum of about 100 W from the mains, and yet is supposed to output 800W to the speakers. Where have they magiced-up the extra 700W from?
5) they sound rubbish. I'll qualify that by saying it's the sort of sound that seems impressive for the first 15-20 minutes, but soon starts to grate on your ears. After a while you'll come to realise that there isn't proper bass. It's just an undefined thud thud thud rather than playing bass notes. Also, the main speakers sound brittle and harsh and lack the ability to convey any sort of midrange detail
It'll be a similar story with any similar <£300 HTiB system, and there aren't many to choose from. The Sony BDVE4100 and the LG LHB645N both offer tall floorstanding front speakers for your £300-£320, but that's pandering to people who look at their audio gear rather than listen to it.
Where I would recommend you look at the Yamaha YHT1840 system (£350).
Here you have a strong home theater brand making a decent AV receiver and packaging it with a good entry-level speaker system. You get very few bells and whistles.
There's no smart features, nor a BD player, and no NFC or Bluetooth or wireless. What you do get is better sound quality and four HDMI inputs with full HDR wide colour gamut 4K pass-through plus HDMI ARC so you can get sound from the TV and its apps in to your surround kit. The amp and the speakers are upgradeable with after-market hardware. They've made a minor compromise with the sub; it's driven from speaker outputs on the amp rather than having its own power amp built in. However, the amp does have an RCA phono socket for connecting a better sub.
edit: typos
