In general with tents as others have said it’s more comfortable if you can stand up. But depends on what type of camping. If you’re hiking and carrying for a long time you want the lightest one you can get which will usually mean it’s quite small/no standing room.
If you’ll always have access to a car then weight is less of an issue but you need to be able to pitch by yourself. Air pole - Vango air beam or the decathlon air ones are great. Generally get a tent that’s meant for one more person than you have so would say for you and the dog that’s a 2-3 person. In your case I would say go for one with an attached fly sheet and without that in wind it’s quite hard to do it solo. Pop ups are also a good option but check weather -rain and wind rating. Although I’ve successfully camping in downpours in Wales in October in a Decathlon pop up. (We’ve never had a bad experience with a decathlon tent and have bought 4 from them).
you can go with all the kit as others have said - we have a 5 person tent (family of 3 plus sometimes dog) and have camping cupboards, two tables, camping chairs, double air mattress plus single mattress pad for dog and one for child, separate kitchen shelter etc - for week long trips with car. Tent weighs 35kg alone. It’s a serious amount of kit but makes for a very comfortable set up. We don’t bother with pillows but take pillow cases and use our clothes in packing cubes inside as pillows which slightly cuts down on bulk.
or we also have a hiking tent where we don’t bring chairs, tiny pocket rocket stove. we have thermarest neo air mattresses (15 years old now, were very ££ when bought but worth the money). You will want somewhere to sit though so think about lightweight stools, waterproof pads or camp where they have stumps or logs or near a pub. This is the set up with used for a bike packing trip for two weeks but we stayed at campsites at least every two days with decent amenities, on site cafes, access to laundry facilities etc. As those trips are all about less stuff and more moving.
So it completely depends on what type of camping so decide that first before you think about kit, but since you’ll be carrying it yourself think about how many trips back and forth you can do with the kit. Start small and you can add to your set up. I would say go for a good quality air mattress that you can use everywhere which isn’t always the thickest. I do recommend thermarest as they are great quality. A thick mattress in a small head height tent = more condensation so I personally would go for a slimmer mattress and pop a foam roll mat underneath for the extra insulation which is the warmest set up. Don’t skimp on the mattress- We had some cheaper self inflating ones which really only lasted a couple of years so were a total waste in the end.
buy at least a three season sleeping bag, as you’ll end up getting one anyway if you ever camp in the UK. Nothing ruins a camping trip like being cold and gets surprisingly cold at night even during summer. And always bring a warm hat and warm socks for nighttime.
And get a stove - MSR pocket rocket is great if you want to keep it light, but a Coleman suitcase type one is easier and more stable. Get nesting pots and pans, you don’t have to bring them all on every trip but you’ll usually want a hot drink in the morning. A kettle is fine on some trips - noodles, instant porridge, instant cappuccino/hot chocolate are all practical camping options for if you plan to eat most meals in pubs and cafes, you have covered breakfast or a quick meal if things don’t work out.
Look out on local groups and online as people often sell old kit at the end of the season when they upgrade or decide camping isn’t for them.
also consider a warm coat for the dog. We discovered our dog was getting cold at night when he reached the age of 7, so bought him a fleecy coat for overnights in tents which made a big difference to him.