I used to get red bumps and sensitive skin after shaving, so I sympathise. What worked for me was the following: -
1. I dumped the gels and supermarket razors, and bought a DE razor. I chose a Mercur Futur (this one:
https://www.amazon.co.uk/Merkur-Futu.../dp/B009ZF8Y18) after reading up on the subject. Generally recommended as a good choice for a "first time DE shaver" since you can adjust the closeness of the shave.
2. I bought a shaving brush and some shaving foam. Tons of choice here, I like Trumpers shaving cream, and the rose seems good for sensitive skin.
3. I was recommended to use face scrub - I discovered that my razor bumps were actually ingrown hairs, so giving a once over with face scrub fixed this for me within a month.
4. Buy a selection of DE blades (most shaving places sell these) and work out which work best for you. Feather blades are my preference, but everyone is different.
5. Learn (or rather, re-learn) how to shave. Sounds obvious, but DE razors are a lot less forgiving than your average Gillette. Take your time, don't press too hard (use the weight of the razor), and don't always try to get a totally smooth shave. When I started with the Mercur, I'd just give a one pass shave, and touch up any awkward areas (under the nose, corner of the mouth) with the old Gillette. That worked for me, and I quickly learned how to shave well (and only) with the Mercur. Shaving in or straight after a shower is great advice.
6. Slap on some aftershave balm. I don't use aftershave, instead I use unscented balm. Cools the skin and helps with any initial redness.
It's a bit of an effort initially - I went from sleepily dragging a Gillette around my face in 2 mins, to actually devoting 10 minutes to doing it properly, but it's just second nature now, and my skin is totally clear now. No irritation, no ingrown hairs, no razor bumps.
Good luck!