Abby,

Thanks for your support and compliments to the MGV program and our efforts to educate the public.   

This link will take you to a detailed summary on lawns and care. Please consider a soil test to determine the current state and conditions before doing anything drastic.   

https://ohioline.osu.edu/factsheet/hyg-4031

For your flower / vegetable garden areas see this. 
https://extension.uga.edu/publications/detail.html?number=C1144

This link will lead you to several options to consider to prevent weeds in you garden.  Garden weeds can spread to lawn areas likewise grasses gone to seed spread into garden spaces. 

I prefer to weed a bit every time I’m in the garden to keep them in check. This is the same when I walk out on my lawn. I take a small pick and dig them out when they are small. 
 

I find if I wait until after I mow or harvest and water my garden I’m often tired or unwilling to spend the necessary time weeding.  The smaller they are the less disturbance pulling them will have on your vegetables and other plants. 

Use of shredded newspaper and Kraft paper along with the suggestions on this link will be ideas for you. Light cardboard from gift boxes, central, cake mix and such can be used as well. I avoid wax coating however. 

A leaf pile accumulated all year round is good as long as seeds are not in them that can germinate and become weeds. 

Enjoy your yard and garden space. 
Barb