We are in the military. We have moved 8 times in 6 years, so of course we rent. Currently, we are in a rental in a pretty nice neighborhood. In our lease we are suppose to care for the lawn and house like it's our own. Caulk the tub, clean the gutters, hire out pest control etc. When I moved in the lawn was pretty sad and sparse. We cut it, water it and rake the leaves but that's it. It looks like crap. Lawn care people stop by all the time offering to fix it up. On one hand I hate that it looks crappy and I feel a little obligation to fix it, on the other hand I don't think it's worth my time and money. What do you think?
I think you might want to check your local landlord-tenant code, which ideally was provided to you by the landlord when you moved in. If not then you might be able to find a copy on your city/county/state website. I'd also start taking "before" photos now so that you have a basis to compare to the "after" photos that you're going to show when it's time for the landlord to return your security deposit.
If you decide that you'd like to improve the lawn then I'd ask the landlord to either pay for the raw materials that you want (grass seed, fertilizer, tools?) or to help with redesigning the landscaping to match the climate (xeriscaping or mulching). I think lawn care people are mostly interested in getting you addicted to the services of Messrs. Mow, Blow, & Go.
How much longer are you expecting to be around this lawn, especially if you live in a climate that has "winter"?
Speaking as a landlord, if I expected tenants to handle the caulk and the gutters then I'd get exactly what I deserved.