If you are not driving, your options are pretty limited I would say. Mears does offer transport from hotels to attractions, see their website
http://www.mearstransportation.com , otherwise you will have to take a taxi, certainly to the malls. To be honest, with 4 of you, I doubt that a taxi would be more expensive than Mears and it would be a lot simpler esp with small children in tow. I don't think taxis wait at your hotel, but ask in reception and they will call you one. The public bus service, Lynx, will be very complicated for you to access, in my opinion, though others may disagree. I've never tried it. We have always had a car, but I sympathise if you don't want to drive as I always make my husband do the driving!
As the children are so young, you may want to leave Universal till next time. There is a young children's area in each park, but the majority of the rides are really aimed at older ones.
If you are having to get a taxi to shop, then the outlet malls at Lake Buena Vista will be nearer than the Florida Mall and therefore presumably a cheaper taxi fare. Tell yourself when paying for taxis that a rental car would have cost you £300 or more in essential extra costs to the 'free' car, so you can spend quite a lot on taxis and it will still work out cheaper!
'Brits Guide' is also a good book, I agree.
You will have to take things gently with young kids at that time of year. Be sure to take a break back at the hotel in the afternoon. Make use of the 'early entry' privilege as with the time difference it is easy to get up early, esp the first week. Save the late stuff like fireworks for the second week, when your body clocks will have adjusted. It is important to always be at a park before it opens, ready to go in as soon as they open the gates. You will get on more rides in that first hour than the rest of the day put together! Understand 'Fastpass' and use it. The website
http://www.allearsnet.com is a very useful one.
You will have a brilliant time I promise and you will be back for more, so don't worry if you can't fit everything in.