Firstly, sorry to hear of your son's illness. I hope things improve for him.
I've not been up Kilimanjaro, but I know it's high enough for altitude sickness. Definitely a good challenge.
I have done stuff in the Alps and Lakes / Scotland etc though. So,
Boots - sounds obvious, but fit is the main consideration. Lots of good makes on the market... Scarpa being one of the best and popular, others like Meindl, Asolo, Brasher,Salomon, etc. For walking in the UK, you'd want 2 - 3 season boots. 4 season will be too stiff unless you want to wear crampons. Either go for full grain leather (which is thick enough to be almost waterproof, or thinner leather (or fabric) with goretex type membrane. You also want Vibram soles, although Salomon and other use their own.
If you buy boots, it's worth breaking them in a bit. Modern boots don't need much, but you don't want to find you get blisters in the middle of the 3 peaks.
Socks. Get some good walking socks. I've always worn 2 layers. Thin socks and then walking socks to prevent blisters, but it's personal preference.
Coat: You want a waterproof, breathable jacket. Go for something with a membrane like Goretex, eVent etc. The ones without (i.e. a treated surface) tend to lose their water proofing after a while. You can pay anything between around £85 - £300. Generally, around £100 - £150 would be good. I'm a fan of Mountain Equipment, Montane, Berghaus. But plenty of other good brands.
Waterproof trousers also important for rain. You can get ex-army goretex (camouflage) for about £30. The "proper" walking ones are expensive.
Mid layer. Some kind of fleece (polyester). Although a wool jumper works just as well (actually warmer).
Base layer. A "wicking" layer, not cotton, that takes the sweat away and stops you getting cold when you stop. Plenty of choice of these. Most running tops are wicking.
Walking trousers. Can get with zip-off legs. Fast drying.
Hat - keep your ears warm. 30% of body heat lost through your head.
Gloves. On the tops of mountains, with wind chill, you will find gloves are important.
Neck tube e.g. buff. Handy for keeping the draughts out and multi purpose.
Rucksack. For day walking, then up to 30 Litres capacity should be fine.
Compass and map. Definitely worth getting someone to show you how to set and follow bearings on a map. Can be a life saver.
Water bottle etc. You can get bladder type containers too e.g. platypus / camelback.
Basic small first aid kit.
Head torch e.g. Petzl. Very useful if you get caught out by fading light.
Alpkit go good value kit. I'm not linked to them, I just have some of their kit.https://www.alpkit.com/