Pete, I had the same issues along with painful ingrown facial hairs as well. The first, and most important step is ditching the modern plastic fantastic 15 bladed wonder razor, and it's the single best change I ever made for my skin. And changing the way I shaved not only virtually eliminated all my issues, it also made look forward to shaving rather than dreading it. That's kind of the mentality behind making the switch, is that you embrace the process rather than hating it as I did. It's like anything else, you can spend as little or as much as you want when getting started. You can use a DE (double edge, also known as a safety razor back in the day) razor with your regular shave cream you use now with no issues and that would likely cut out a lot of your problems.
If you choose to take the full plunge however, or if you simply desire to give it a try then a decent bowl and brush, along with shave soap can be picked up cheap online. The brush is key also, as it exfoliates the skin getting rid of dead skin helping the razor glide along easier. In addition it softens and lifts the whiskers prior to being cut as well.
My current setup (tried and true for about four years now)
- 1970 Gillete tech adjustable DE razor (adjustable gap for the perfect shave)
- Derby blades (you can buy 100 of them for roughly the cost of 5 plastic fantastic cartridges)
- Escali badger brush (very reasonably priced and pure badger hair)
- Proraso eucalyptus & menthol shave soap
- Cheap stoneware bowl from Wally world
Another valuable tip my barber gave me was to never immerse your razor in really hot water or run it under a really hot tap, as it tends to warp the blade and change the gap on the adjustment setting. And when finished shaving, rather than splash cool water on your face, soak a hand towel in cold water then apply it to your face and neck area with a little pressure and repeat if it warms too quickly. This helps the pores shrink (they don't truly open and close, that's a myth) and is healthy for your skin.