My own two pen'orth.
Without knowing how to use them, axe and crampons may be more hazard than help. i.e they may give you the impression that you have a safety factor and tempt you to go a little further than you would be comfortable with otherwise.
Althought Wayne makes an excellent point that it's very un-PC to recommend that someone doesn't take a piece of safety kit, it's skills and knowledge that come first. I despair of some of the walkers I've seen out and about the lakes playing about with long walking axes in two inches of snow on some tiny incline and totally messing up their balance and posture. They're asking for an accident !
An axe on it own is next to no use on ice. If you expect ice, then you need crampons and some practice walking in them so as not to trip over your own feet. If you do have the crampons, then on anything shallow, the axe won''t have anything to do. Stay upright and relaxed, think about good balance and just walk. The summer walk up to the top of snowdon from the car park at pen-y pass via the lakes is a case in point. Training shoes in summer, but the path covers in sheets of water ice if it's cold/ windy in winter. It you don't have spikes, you've no business trying to continue up this route.
Based on the two points above, my own view is that walking axes are often not the best tool. If I expect conditions may need them, I always take crampons and axe together, and the axe is a long (50cm) climbing axe. This is far shorter than a walking axe. There's no temptation to use it like a walking stick on gentle slopes, but when the slope is steep enough that an axe starts to be needed, the shorter length is much easier to handle. It has a full sized head so still makes a good brake. I'd be interested in the opinion of the experienced guys on that choice ...
Oh, and I totally agree with mmckniven, forget all this step cutting nonsense. It's hard work and a real skill that takes time to learn. If it's steep enough that you need to do this, you should be using crampons.
On soft snow, you can stomp around without campons much of the time and if it's not too steep and there's nothing to hit, I wouldn't let lack of these two tools stop me going out - I would be plan a route where I had time to to turn around and retrace my path if I met conditions that I couldn't deal with.
As you haven't done it before, the idea of renting an axe and some crampons and going out specifically to learn how to use them is excellent. It's good fun and as well as boosting your confidence when you've cracked it, it will teach you some of the limitations. Quick reactions are important to stopping a slide early before you build up speed. Once you're moving, your stopping distance can be rather long. A good analogy is stopping a car when it starts to skid. Knowing the theory isn't enough, you actually need practice to be able to do it.
Definately go with someone else, preferebly someone who can teach you, and try all the variations. You have to be able to stop all four slides - all combinations of feet first and head first, sliding on your stomach and back. Seems to me that it's usually the two head first falls that are more likely to get you. And don't cheat ! Head first on your stomach is a lot harder when you pack rides up, smacks the back of your head / helmet and stuffs your face in the snow
):
Then do it with the crampons on so you know what happens when you go feet first, they catch and flip you head first :wave:
When you start to learn, you'll realise (maybe with some surprise) that you need what feels like a pretty steep slope to be able to slide at all, and how big a difference a smooth pair of waterproof pants / long jacket makes compared to more "wooly" clothes.
Be particularly careful about the thickness of snow and the potential for rocks. When I was much younger I had to accompany a very pretty and very embarassed lass to the local clinic to have repairs made to her bum after we failed to take enough care over this :yikes:
I think you have a great opportunity here - go for it :biggthump
:embarasse and slightly flattered 
