Also wondering about carrying stuff.
Depends on how much you want to carry, and your preferences.
I find it hard to steer with a front basket, the whole thing is too front-loaded. Cycling with a backpack is the least-faff option when locking up. It doesn't require any other expense or weight, e.g. basket, rack or panniers. For some reason though cycling with a backpack gives me backache.
I have a rear basket on a shopper bike. The bike has small-ish wheels, so it is still stable even when loaded with groceries, and really convenient. A rear basket on a full-sized-wheel bike would be less stable if loaded up, but again it would be fine for some, e.g. plenty of people cycle with a toddler on the back.
I like panniers because then the weight is low, but they are a faff when locking up, and less comfortable to carry about compared to a back pack. You'll need somewhere for your lock, which can be really heavy, some bikes have geometries which don't allow a D-lock to attach to the frame.
Are you near a city which has a bike rental scheme? Or somewhere touristy or picturesque which might have a bike rental shop? The latter are likely to be mountain bikes, which will be heavy, but mountain bikes have a large range of low gears so you'd be able to try out what you need for your hill.
What bikes do your children have? What do they like/dislike, would it apply to you? How easily can they cycle up the hill?
If you can't try anything out, or buy a cheap second hand bike, I'd go to your local bike shop, explain your requirements, compare the weights of any bikes that they recommend, insist on a small front chainring, and if possible, handlebars which can be raised. Make sure that there are attachment points for a pannier rack and water bottle holder, and space for a D-lock, in case you decide later that you want these.
If the hill turns out to be too much, but the bike is otherwise good and fairly standard in construction (e.g. not carbon fibre), and your budget can stretch, you could have a battery + motor retrofitted. I used Electric Bike Conversions.