Consider, bluegrass and creeping red fescue mix. Bluegrass will do better in the sun and the fescue better in shade. Kill weeds before planting seed. Then add any need nutrients and rototil in. Level and remove clods by raking. Then seed and lightly roll. Need to still be able to leave foot prints in soil after rolling - if not the roller is too heavy. water as need to keep grass seed and new seedlings alive. Once grass is growing, water less and less often but deeper and deeper till you can go 7 to 10 days between waterting. Do not try to control weeds that come up till late summer early fall.
Do a soil test for your lawn and another one for your vegetable garden area. Once you get the recommendations back - apply the needed nutrients and rototil in. Make sure vegetable garden is in full sun. If first garden start with a small one such as 10 by 10 feet. If you stick with it - you can always enlarge the garden in future years. Many people give up vegetable gardening because their first garden is too big and too much work for a beginner - hence the suggestion to start small. Also consider applying any needed soil amendments to increase drainage or to retain moisture based on how compact or sandy the soil is.