I use my credit union's online bill pay, but dont' do them automatically, I schedule the payments manually. There's no fee from the credit union to do this.
As far as the bills themselves, I toss the outer and inner envelopes, and I have a cheapo monthly calendar "planner", and I shove the bill into the appropriate month's pages. I have all the due dates pre-written (I do 3-4 months at a time) in on the calendar (it's also where I track one-time misc bills, appointments, bdays, and other stuff) so I just check it once a week or so and then pay whatever needs to be paid. I can't remember the last time I wrote out a check, but when I did paper checks, I did essenaitlly the same thing but kept the outer envelopes.
Once bills are paid, I write the date, amount paid, and electronic confirmation number on the bill (used to write the check # when I did paper checks), and file it in an accordian file. The accordian file is for the current year's bills, and then anything that isn't tossed after a year goes into our big filing cabinet, until it's time to toss that.
I would highly suggest looking into an online bill pay feature through your bank if I were you, though I don't know that I'd pay a fee for it, necessarily - unless that monthly fee would be the same or less than I'd be paying for postage to mail out the payments anyway. Most online bill pays I think you can still do manually if you want, as opposed to auto withdrawls...auto withdrawls still freak me out for some reason, I like to have control. The only thing we have on auto withdrawl is our insurances, as we get a discount doing that.