Hi Bill - I've used a number of different property management software systems in my career as a professional property manager.
Yardi - Way too cumbersome and expensive for your needs. Requires hours and hours of training just to get it be functional for you. It is primarily property management only and can't handle your other needs to run credit etc. The accounting module will make your eyes bleed since it is way to complex.
Buildium - Folks seem to enjoy Buildium since they are going from spreadsheets to their first management software. This can blow you away, if only because working with spreadsheets is a nightmare way to spend your time as a real estate person. It is very limited on the accounting side but for your first experience, it may be just right. For those of you with lots of properties, you may hit a limit with Buildium and start to look elsewhere for something more robust. Migrating your data to a newer platform is a terrible and disruptive transition. Buildium is not that customizable if you need something outside the box. Which, for me, defines 99% of real estate.
Rent Manager - My VP suggested we use Rent Manager for our office because it is very customizable. We have not looked back since. Basically, if you can get over the sheer amount of customization and features it offers you will likely not need another piece of software until you or rent manager goes out of business.It can handle auto payments, credit checks as add-on modules you can add for a small fee. I've heard of cases where people have left Rent Manager for Appfolio but I can't speak to that experience. Hopefully someone can chip in with their Appfolio experience.
I'm not going to quote pricing on these because they all offer different schemes for payment and it depends on how you get it delivered and the number of licenses needed.