Hmm...what is your budget for monthly rent? With Mather's central location, you do have some nice options in terms of nearby areas that will be a manageable commute with the amenities you are seeking. Mather itself (95655) does have its own communities nearby. There is a good amount of newer construction there if that is something that appeals to you. There are also a number of older communities - mainly ranch style homes constructed in the 1960's and 1970's.
I have two areas in Sacramento that are personal favorites and have the amenities that you describe...rents in the two following areas will likely be on the higher end, and you will probably find the areas in and immediately surrounding Mather to be in a more moderate rental price range.
I am partial to the Land Park (in 95818) neighborhood, although I do not live there myself. It is bound roughly by Broadway to the north, Sutterville/William Land Park to the south, I-5 to the west, Freeport to the east. Elizabeth is correct when she describes the neighborhood amenities there. There is a great selection of public K-12 (Crocker Riverside Elementary, California Middle School, CK McClatchy High School) all within the confines of the neighborhood. William Land Park is a very large park at the southern section of the neighborhood that has a great duck pond, picnic areas, a golf course, a small outdoor ampitheatre, a children's amusement park (Fairytale Town), the Sacramento Zoo, etc. There are public events at the park all the time. The "walkability" of the area is great...and if you want easy access to freeways, I-5, I-80, Hwy 99, and Hwy 50 are all very close. Public transportation access is also very close, and downtown Sacramento is close if that is important to you. Your commute will be in the opposite direction from most commute traffic, however you will be a major interchange and several freeway exits from Mather. The majority of the homes there were built in the 30's, 40's and 50's. They are very charming and many have some neat architectural characteristics.
Another great area in a different part of town is Arden Park (in 95864). It is bound roughly by Arden to the north, Fair Oaks Blvd to the south, Watt to the west, Eastern to the east. While these homes were built in a similar era as the homes in Land Park, architecturally speaking they are not as unique as those in Land Park...these are mostly ranch style homes on larger lots (between .25ac and .75ac-ish). There is also a great selection of public schools there, however only K-8 (Mariemont Elementary, Arden Middle School) within the confines of the neighborhood...since you have younger children - this may be just fine for you? There are two very nice parks in Arden Park - Cresta Park and La Sierra Park. Arden Park actually has its own recreation district, its own tree foundation, and lots of neighborhood-sponsored events. The walkability here is also good, although the neighborhood is fairly spread out and not as dense as Land Park. The closest freeway is Hwy 50. Driving to Mather on a daily basis, you would be only 2 freeway exits on Hwy 50 from the Mather Field exit...commute traffic for the most part will be headed the other direction (you would be heading east, most commute traffic is headed west).
Either way, welcome to Sacramento, and I hope you and your family are very happy here.