Here in Illinois, the rental market is very competitive. So, my first tip would be, don't dally around too long if you like a place. Also, in our area, rentals are averaging about $1/sq. ft per month. Not everywhere but, it's a good rule of thumb to keep in mind when deciding to negotiate a rental price.
For the lease, I always write in that the owner maintain a home warranty. It makes the owner's and tenant's life easier when it comes to repairs to the home.
For tenants, owners are still wanting good credit scores but, the credit report tells the story. Maybe your scores aren't the best but, if the report shows good, recent payment history, it helps soften the bankruptcy or foreclosure or charge offs that may have killed your score.
If your score isn't the best, the owner may ask for a larger security deposit - 2-6 months rent.
I'm sure there will be many other good tips provided by my colleagues.