Lisa - will let someone more knowledgeable give you a proper answer on that, only thing I will say is that walking to recover is definitely fine in my book, personally I just drop back to a slow jog but my super-speedy friend does all his speedwork on a track and actually stops completely between reps and swears this is the best method (don't like the sound of it too much though, sounds a bit jarring..)
I think if you can, idea is to do a few reps to start with (I did my first speed session in months and months today, and did 5 x 800m) and then try and build them up, one more each time maybe. But there are loads of different ways you can do it - increase the length of the rep, reduce the recovery time, add more reps etc. Not really sure of the 'best' as I'm no expert on speedwork.
Congrats on your run tonight too by the way Lisa! Speed is all relative to yourself and sounds like you have improved a huge amount, must feel good!
Anyway my MAIN point - [sorry everyone, I do tend to waffle!] is -
I used to go to a gym and do my speed sessions on the treadmill as it was easy to control speed but since I've switched to doing them outside this winter with the rest of my runs I've got a lot more satisfaction out of them. I generally hate running fast but when they are short bursts, but just long enough to get a nice pace going, they feel fantastic! Definitely definitely feel a lot more rewarding doing them outside so would recommend getting off that treadmill if you can! Although looking at your watch if you are doing them by time is annoying, luckily I got a Garmin for xmas so don't have to worry about that anymore