The DFT sends out Tenders for prospective train operating companies to bid on. These used to be funded by mixtures of subsidy/fares (or fares-repayment), but now they are fixed price contracts to run the services.
The good news is, you don't need to buy any trains, they are leased from ROSOCOs (rolling stock operating companies), who own the trains.
You'll then need to pay track access charges to Network Rail to use the track.
The bad news is, well, you can't really do much in terms of changes to services. The DfT has certain specifications they are looking for i.e service patterns and the network is very congested, so finding room to run any extra services is difficult.
Open Acess operators are free to do what they want, but you won't get any money from government, so they are limited to only the most profitable, in demand of routes that existing TOCs don't cover (so not many). Hull trains is one, Grand Central is another. Lead times on new rolling stock can be up to a couple of years, so finding old rolling stock is probably a more likely option. Then there's the costly process of making it accessible, etc. You're also stuck battling it out for slots with existing TOCs, which is a challenge, especially around the UK's major cities, most of the infrastructure is running at over 100% of reasonable capacity.
Expect the whole process to cost at minimum £10 million, very likely higher, especially if you wanted to go the open acess route.
If you've got some billions to spend, and it's really burning a hole in your pocket, let me reccomend helping out the fine folks of Leeds on building a tram system. God knows they need it!