How would I do things such as splitting wood etc with just things such as a knife and saw etc
As long as you have a decent knife (and can use it safely) By decent knife I mean something like a Frosts Mora; you can carve out a bow with no problem.
Don't worry about chopping down trees or splitting staves. All you need is a reasonably straight Hazel branch 1 1/2 - 2" thick by 5ft long. While it is still green (ie when you have just cut it) remove the bark by scraping the knife along the length of the stave, taking care not to dig into the actual wood.
When you have all the bark removed you need to find the centre of the bow and make a pencil mark. Then make 2 more marks 3" either side of the centre mark (this will be your handle and will remain full thickness when drying out.
You now need to look at the stave and decide which is the "back" and which is the "belly". The less knots on the back the better as this is where the bow will break when under tension (drawn).
With your very sharp knife (and First Aid Kit to hand) you need to reduce the belly side of the bow until it is half the thickness you start with. Make sure you take the wood off the same on both limbs (the stave either side of your handle). At this point do not attempt to bend the bow or make the limbs narrower.
Then put it to one side on a flat surface and leave for 2-3 weeks in your bedroom under the bed (or wherever) to dry out.
When you get this far I can post the next installment...
