I have set flyers and business cards around my neighborhood about lawn care services. Now, how much do I charge to cut their yards?

Good move about the flyers and business cards. You might continue to do that on the slow days, moivng further and further out in your area too until there aren't any slow days!

Tough to know what to charge when you don't know what the competition is charging. One way is to offer to do it for 10% less than the competition. You can state it so they have to show proof if you are concerned about honesty -- "Use your last lawn service invoice as a 10% off coupon with us!" Make it the reduced price for some time too. Firmly build your relationship with your customers and down the road, raise prices.

Or the other thought I have is to ask about what your competition charges. Pretend to be a potential customer and call them, or ask about rates at the mower place or garden center or even a real estate office -- people who regualrly deal with lawn service folks may know.

Everything can depend,

In Atlanta for example, the law services charge about 40 dollars to do a small yard that can be done in about 30 minutes. In Rural Tn I pay 50 dollars for them to mow over a arce yard and then about 25 dollars for them to do a very small yard.

I could get it all done about 15 dollars cheaper for both with some other companies even but the one I use provides me great service, so I personally don't want to change.

I would guess you have a "idea" about how long it takes to do different yards by doing yours at least, and you know the type of equipment you have. So you estimate.