Sluni,
There are only 7 towns in the entire state of NH that have more than 25,000 people, so you have plenty of options to choose from.
Your interest in walkability makes me think that you would
really prefer the towns in the Seacoast region - these are the oldest towns in the state and have dense, vibrant, historical, and walkable downtown cores. At the same time, they maintain that peaceful, quiet, leafy green atmosphere that makes NH such a desirable place to live. Oh, and safety is almost entirely a non-issue when talking about places to live in this state - seriously.
Portsmouth, of course, fits this description to a T - it's on the larger end of your size requirement, but it really is an exciting little city with a decent array of cultural amenities for its size. The other towns in the Seacoast region that fit your needs are only a short drive from Portsmouth - Durham, Dover, Exeter, Newmarket... and these towns all have cultural offerings of their own. Durham is home to the University of New Hampshire, while Exeter is home to the renowned Phillips Exeter Academy. These two towns, in particular, are considered as having two of the very best school systems in the state. But the other towns have good to excellent schools as well.
These towns are all under an hour's drive from the Manchester airport, and just over an hour to Logan Airport in Boston. Dover, Durham, and Exeter all have train stations in the middle of town that can take you to downtown Boston, and Portsmouth has frequent bus service to Boston.
Of course, there are also some fantastic school systems in the Merrimack Valley region, closer to I-93, but I would argue that most of the suburbs of Nashua and Manchester are definitely less pedestrian-friendly than the Seacoast towns. Places like Amherst consist largely of autocentric suburban development. These towns tend to have larger, more spread-out populations and feel more like bedroom communities than cultural centers of their own. That's my general feeling, anyway.
Either way, you're going to pay a premium to live in a town in southern NH with great schools - whether through property values, property taxes, or both. If you gave us an idea of your budget, maybe we could help narrow down the choices.
The Seacoast definitely gets snow in the winter, although it's usually much less than the more interior parts of the state. However, the mitigating effects of the ocean air disappear pretty rapidly as you head inland - it can be surprising sometimes how much snow will fall in Dover versus Portsmouth. I think Dover actually gets a bit more snow than Nashua on average.
It's about an hour and a half drive between Portsmouth and the edge of the White Mountains (near Conway), and I would allow two and a half hours of driving (NOT including stops) to make the scenic loop through the mountains. So for a day trip from Portsmouth, it involves a lot of time in the car (about 5 and a half hours) - which is why it may be worthwhile to spend a night in the mountains too.