Straw might clog up the tiller if it isn't rotted down or chopped. You might consider loading up with mulch at the end of the season and using a push mower to chop it up. Or you can put your mulch on now and let it rot down.

We used to use a tow behind mower, too. I sold it on Friday because we are mowing a much smaller yard now and bush hogging once a year for the rest.

Grass clippings will work as mulch, but you have to put it on in thin layers. If you put it on thick it will get very moldy. The mold is nasty even if you aren't allergic to it. It helps if you let the clippings dry out before you rake it up. Fresh grass clippings will turn moldy in just a few days.