Well, I lived in Eastleigh for many many years until I went overseas with my husbands job.
I still own in Eastleigh, so can give you a good idea.
If you do want Eastleigh, buy Boyatt Wood area. Good Infant, Junior and Senior School within walking distance. You also have the rail links 15 mins walk (if you go over the railway bridge), a good bus service, and an airport down the road. Close to M3 motorway, and the M27.
In town there is a good park for the children that has just been refurbished, and when I was back in September it looked pretty decent.
There is a shopping centre there, with a library and the usual Argos, sports shops, clothes shops etc.
In the High Street you have most of the banks. Barclays/lloyds/Natwest/HSBC/Santander. The High Street also has many of the Estate Agents. Plus the food shops (Tesco & Sainsbury).
There are plenty of pubs and restaurants in Eastleigh town ctr too.
Locally I would say Colden Common is a little more rural, but I cannot comment on the schools there. It is a bit more expensive, property wise, and you would probably need a car to go food shopping etc.
Shawford is lovely but very expensive, same with Twyford.
Chandlersford is nice in areas (much like Eastleigh). I'd say Valley Park, and Hursley are very nice. Chilworth is lovely too. Chilworth is extremely pricey though.
I'd avoid the Eastleigh side of Southampton. Whilst it is cheap, it is cheap for a reason.
Any questions feel free to ask. My parents and grandparents still live in the area so anything I am unsure of I can check.