Fellowships can be divided into two categories: "Senior" and "Junior".
The senior fellowships are the most prestigious; they usually last for 5 years, give you significant amounts of money for equipment/consumables and you are effectively an academic. However, you do need to have quite a few years' postdoctoral experience under your belt (or have held a junior fellowship) before you become competitive for them. The work involved in applying for these is generally quite a lot and you will need to get involved with the financial side of things along the way.
The junior fellowships are less prestigious; they usually last 2-3 years (some may not even cover your entire salary), they give you less money for consumables and you're not really an independent academic because you don't have the money to be treated as such. Junior fellows are usually attached to an established research group - you will be carrying out your own research, but instead of operating completely independently, you will be heavily reliant on equipment and cost sharing with an established academic. However, they are much more straightforward to apply for - typically a 2 page scientific proposal and an application form - and you can apply for these straight out of your Ph.D. if you so wish.
If you're unsure, the best thing you can do is to try and write a 2 page scientific proposal (A4 size, including references) outlining your idea and how you intend to carry out your research. This can be expanded if necessary, but most of the junior fellowships only need a 2 page scientific proposal.