Lived here for many years and my kids get the trains out of the city everyday for college and school so like you we need to be a short walk from a station. Your options are.
Portsmouth Harbour - Gunwharf Quays to one side, over priced apartments and over crowded on the weekends. Portsea to the other side, an area best avoided even though a few nice blocks have gone up in recent years.
Portsmouth and Southsea - Next to the University, anything affordable already a buy-to-let with students, the rest is social housing. The main shopping centre (Commercial road), we only go there if we have no choice.
Fratton - This station is walking distance for homes in Southsea, Eastney, Fratton and Milton. However, the majority are terraced houses, roads are congested, difficult to park and lots of students as the areas are serviced by a free Uni bus.
Hilsea - Bigger and more affordable homes but too close to the motorway, noise and air pollution. And not all trains stop at Hilsea.
Cosham - Some nice homes towards Drayton, not too sure what the area is like now. It's close to Paulsgrove which doesn't have a good rep.
If you can find somewhere nice to live in Southsea close to the common then it's a nice place to bring up a family. Lots of free events on the common throughout the year, the sea front is a nice place for walk or run with nice clean and safe parks, play centres and cafes.
We're a short walk from what's considered the best primary school in Portsmouth and both of us have a 10 min walk to work. My wife doesn't drive and I rarely use the car. Shops, schools, work and pubs all a short walk but we'll need to move soon as we need a larger place and I'm not paying stupid money to live next to students. Just waiting for the youngest to finish primary school and we'll move out, somewhere between the A3 and M3.