I wouldn't fertilise the lawn as of yet, as it might burn the new grass... give it time to come up & get stronger... for patchy areas, then add more grass seed. When mowing, keep it at a high cut level - should only be taking off the very tip of the grass & don't mow if the grass is anyway damp. Avoid walking on the grass / soil for as long as possible... just leave nature kick in (watch out for those weeds... they'll be popping up very soon, if not already) - if you are ok with a different "green", then they can add to the effect of a garden - if you don't then make sure you get the root of the weeds if you are taking them out... regular mowing in the growing season (never more than 1/3 to be taken off in any one mowing, at that growing time) will help the grass thickness coupled with some feeding in early spring will also strengthen the root system...
Depending on where you are in the country at the moment... make sure you water your grass if there hasn't been rain in the last few days...