not as easy as you think, you are best off with horizontal dropouts, which i bet 100quid your old bike dont have.
the harder you pull on your cranks, the more you will pull the wheel towards you, look at the dropouts, your wheel will fall off after too many hard cranks. you can counter this with modified chain tensioners, which you will prolly need anyway to get your chain to stay at the correct tension when you tighten your rear wheel.
i wouldnt recommend messing with it unless you know exactly what you are doing, have all required parts and tools (tools is nothing special) have a lot of patience and possibly money for when things go wrong. for instance your rear gear sprocket is a certain width that if you remove, you will need to fill with spacers, or you whole rear triangle, seat and chain stays, of your frame will bend as you tighten it.
fixed gear is for the freaks only imo, i know lots think its cool, but you are way more likely to die on the roads unless you are expert on your bike, like an extension of you or something.
single speed is nice in a way, for exercise, less wires on your bike, less things to break, and it will feel smoother too, (well maybe) and quieter because your deraileur is not clattering about
maybe you can post pics of your bike in its currewnt state?