You buy single tickets on the bus. And returns too - that's cheaper and for the return trip, keep your ticket and show it to the driver. Sometimes even a weekly ticket is purchaseable on the bus.
It all depends on the actual bus operator and in Manchester there are several. http://www.tfgm.com/Pages/default.aspx will help you. http://www.wymetro.com/ is the place to look for Leeds bus information.
For trains, for a short distance buy your ticket at the station before you go. For long journeys, it's cheaper to book in advance on the train company's web site and you can collect your ticket at the station. http://www.nationalrail.co.uk/ is the place to start but always check with the actual train operator's web site for the cheapest fares.
Train tickets can be bought online, ro at the station before you travel (either from the ticket office or from a ticket machine). Bus tickets are usually bought from the driver as you get on the bus - though you may be able to buy weekly tickets from an office at the bus station.
Train tickets can be bought at the station ticket office just before boarding the train, but for longer journeys such as London Euston to Manchester for example, you will find it cheaper to buy them well in advance for specifically timed trains. Booking opens 11 weeks before your journey
www.virgintrains.co.uk To collect your pre=booked tickets from the departure station you will need the same card you used online and your booking reference number
Another alternative if you are planning to do quite a lot of rail travel i Britain, is to buy a Britrail Pass before arrival www.britrail.com
You will find full schedule information for the entire rail network in Britain at www.nationalrail.co.uk
No need to buy tickets in advance for a journey from Manchester to Leeds which is quite short. Most rail services here operate very frequently
On bus anything up to a monthly even possible....In Greater Manchester beware as several operators and if want use ALL need a ticket for all which possible, but only up to a weekly..Otherwise at bus station or some small shops....