I love in Linden Park and rent a very nice 3 bed for $450 pw. If you want to rent in the nice Eastern Suburbs you list then you may have to be thinking more around $350 min per week. My knowledge of Parkside and Unley area is not as good but they will probably be the same price or even higher.
The Eastern suburbs are just wonderful. Excellent primary schools, Linden Park primary has an excellent reputation as does Burnside and Marryatville. so many parks and playgrounds including the amazing Hazelwood Park which has the public swimming pool. The Burnside library is fantastic and as for the shopping......well between Norwood Parade and Burnside Village I am lucky to get through the month without blowing the credit card. You are also super close to the CBD.
as the above poster said, if $300 is all you can go then you might be in slightly further out suburbs. In the east payehnam, tranmere, st morris, glynde area is good and the Trinity Gardens primary school is exceptional.
I cant help you with public transport as I never use it however i see enough buses on the road in this area so therre must be plenty! You will find you drive a lot as Adelaide is an easy driving city
I absolutely love the eastern suburbs of Adelaide and wouldnt live anywhere else but everyone has their favourite spots!
Good luck, adelaide is a wonderful city to bring up kids and you will mind a great lifestyle waiting for you all.