You'd need a big van for 3 adults and a baby who won't be a baby for much longer. Would you have room to keep a van say 6+ metres long and 2 metres wide?
We have a 4-berth van for 2 of us and a small dog. Two of the berths are a double bed over the cab, we use that space for storage, eg somewhere to put the bedding during the day, the table and chairs we use outside etc.
Would you be touring or going to a site and staying there? If the latter, I'd say you'd be better off with a caravan. If you camp on sites with a motorhome/camper, you have to pack everything away every time you go somewhere, and caravans are much cheaper to run and maintain.
You'd also need to consider how the dog and baby would travel. Our van only has side-facing seats, so no seatbelts in the back. We can't take children or secure the dog with a harness. Luckily, we don't want to take children away with us, and the dog's cage fits under the table!
Some people selling vans have "habitation checks" done, that cover things like damp, heating, hot water, gas safety etc. It's worth looking our for one with a habitation certificate.
Things that I would gladly pay a bit extra for are a Gaslow system (you fill up with LPG at petrol stations, rather than faffing about with gas bottles) and a diesel space heater system.
The forum at ukcampsite.co.uk has a motorhome section, there are some very knowledgeable people on there. It might be worth joining.
If you do get one, I'd recommend joining Britstops - places you can stop overnight for free. We've stayed at some fab places and never use sites now.