Wow, your asking me to give u all my secrets/ Here goes. I fertilize in September and then in October. No later. Doing an October fertilizer allows you to skip an early spring fertilizer. With ground cold in the spring, no fertilizer is needed. Watch the lawn companies and they fertilize early and your neighbors are mowing twice a week. If you desire you can fertilize in late April or mid may with half dosage.
I do not use a fungicide. I feel its expensive and if the fungus kills the grass the healthy grass will fill in and you can also seed the yard in September and October. No later.
Grubs and other insects i apply a grub and insect killer in the month of July. Most will need watered in so i watch the skies.I do this every year and no longer have grub problems. I only use one application a year.
Weeds are tricky. I do apply a weed killer in the fall. I use a sprayer and spot spray the weeds. Spring spraying is tricky as the drift can kill your young plants that are growing nearby. Fall, the plants are tougher and the weeds are preparing to absorb nutrients to go into the winter.
Coring a yard is optional. I know it is recommended if you core you should core every year, so i do not core. Your call. Mowing is important. I only bag my grass in the first mowing to heat up my compost pile. otherwise the clippings will fertilize your yard and do not cause thatch build up.
Let me know if you have questions. Glad to help.