To expand on what others have correctly said,
First off, you would need to be a licensed PSV Operator (or in some cases, licensed hackney carriage proprietor / private hire vehicle operator) to get awarded any contract. PSV licensing is handled by the traffic commissioners - if you're not already licensed, I suggest a web search for leaflet PSV437 which is a 58 page document. Hackney carriage / private hire vehicle licensing is done by local councils (districts / city / boroughs where there is still a county or in metropolitan areas - only exception is London where like most things, it's different and part of TfL.) There are broad national guidelines on HC / PHV licensing, but local councils have a lot of discretion when it comes to exactly what standards and procedures they adopt. Some taxi licensing authorities have a special class of PHV licence for vehicles only used on school contracts.
Contracts for home to school transport / social services transport are handled by the relevant departments of county or unitary councils (some such councils have one central transport team who handle local bus, education and social services transport, some don't.)
Generally speaking, they will want some pre-qualification information (e.g. evidence of the licence/s, insurance, some information about your maintenance and management arrangements and so on - e.g. if you're a "one man and his bus" operation, then what are you going to do if you're ill or the bus breaks down?)
Drivers for this sort of work are likely to require an additional level of police checking. Contracts will go through some sort of tendering process, this can be as basic as ringing round for quotes for something to start in a hurry, but this tends more to be at the taxi end of things than buses.
Local bus contracts are handled by county / unitary councils and PTEs (again, in London, it's TfL). There are some forms of bus service that can be operated with hackney carriage / private hire vehicles - this tends towards the rural / demand responsive sort of thing, but not exclusively so.
Anyone can ask to be sent copies of tender documentation, but there is likely to be some form of prequalification process (as above) before you're taken at all seriously. Councils are not allowed to 'discriminate' against specific operators / types of operators, but are allowed to have quality thresholds, and may legally not award contracts to any operator who doesn't hold the right licences.
Councils are under no obligation to give operator X a contract because he asks for one. The 1985 Transport Act (as amended) requires most contracts to go out to formal tender, although there can be emergency contracts (e.g. because an existing operator has ceased trading) and "de minimis" contracts (i.e. small value contracts that simply aren't worth the admin of a tender - although these tend to be small add-ons to existing commercial services.)
Tendering is not always "cheapest price for any old crud heap with a PSV licence" which tended to be many councils' interpretation of the 1985 Act in its early days. Councils can and often do have quality criteria, e.g. maximum vehicle age, emissions standards, and increasingly, low floor buses are specified as standard.
Contracts can be either on basis of operator keeps revenue and price per month from council is fixed; or on the basis that operator income each month is fixed, hence the operator gets tender price less on-bus revenue. In some cases, there can be a revenue sharing scheme which is somewhere between the two. On some tendered routes (e.g. evening services on what's otherwise a commercial route) the tendered operator is required to accept the daytime operator's return and / or multi journey tickets.
Fares will usually be set - either in detail, or in accordance with broad guidelines, by the council doing the tendering. Some sort of electronic ticket machine is usually required - in some cases this will be provided by the council, in other cases the operator is expected to supply this. Some councils do the publicity and maintain bus stops, others expect operators to do this.
Again, usually, operators provide and maintain the vehicle, employ staff and so on all as part of the costs they need to recoup via the tender price. Some councils have some contracts where they own vehicles and lease them to operators; some education / social services authorities employ passenger attendants who then get allocated to runs, some expect operators to employ such staff where needed.
Be aware that in any sort of contract environment, there will be penalties if an operator fails to run in accordance with the contract - this may be financial penalties (you're unlikely to get paid for any journeys you drop entirely) or it may be a 'penalty points' system - and the ultimate sanction is early termination of contract and being barred from re-bidding, either if you do something outrageous, or do something minor often enough.
Ultimately, it is important to read the contract conditions / specifications very carefully before quoting a price, and to allow a bit for 'contingencies' (e.g. dealing with breakdowns etc.)
Any contract is also really only as good as the early termination clause - there will usually be the right for either operator or council to give X period of notice, some councils terminate contracts and re-tender more frequently than others in the hope of driving costs down, and bearing in mind the financial pressure on local authorities at the moment, every year often brings the need to find 'savings' i.e. cuts which means that the bus contracts that have the worst value (using whatever criteria the council chooses) will get the chop.
Although it would be quite fun!