To start with, you should consider dethatching and aerating the lawn. Keeping things simple, get a DIY fertilizer program from your garden centre. Garden Gallery, Sheridans. Scotts, all have recommended programs and timetables for lawn maintenace through the season.
Each component works on different elements of the lawn during the season. Lawn starter, weed and feed, root developer, winter feed, etc...
As for the grass growing past the border. Grass spreads by sending out root colonies. You can never eliminate it, only control. Reccomendations depend upon local spray regulations. There are lots of name brand contact weed killers. If they are banned in your area, the controls include digging them out, bleach, vinegar.
I could not make more specific suggestions without actually seeing the condition of the sod.