I think I need to go this route, or at least give it a try.
Any recommendations on which brand to go for?
If you go into Boots, you should find the Wilkinson Sword Classic for about a fiver and it comes with 5 blades. It is a cheap plastic DE but it works well in practice and WS are decent brand of blades IMO. It's a good starter kit as you're not out too much money if you don't like it.
Edwin Jagger razors are highly rated and should be easily available online if you fancy something higher quality.
Otherwise, find a vintage Gillette razor at a car bootsale, Ebay or antique shop and give it a good clean.
In reality, there are hundreds of different brands of DE razors out there and everyone who uses one will swear by a particular type. I use a 1960s Gillette Tech mostly.