Use Google Maps! I use it all the time when I'm in cities I'm not used to. It'll tell you what exact route to take, what number the bus is, where the bus stop is, how long it'll take, etc etc. Honestly, everyone feels like this when they're first starting out, you're just starting a little later than most. I remember feeling the same kinds of anxiety when I first started taking the bus when I was 12/13, and I've moved around a lot by myself now, and I don't feel scared about public transport anymore, even in cities abroad. Just remember, if you managed to get a bus THERE, there will definitely be one going back home haha. It's basically impossible that you'll get so lost that you don't know where you are.
I don't live in Manchester, but after some googling, I found the 'Young Person Bus Saver' which is about ~£13 for 7 days and gets you on any bus in the Greater Manchester area without having to buy individual tickets. Might be worth looking into if you're going to be a uni student after the summer. (There's a student one that's the same price).
Here:
http://www.systemonetravelcards.co.uk/travelcards/details/young-person-bus-saver-weeklyFor trains and trams btw, there will most likely be ticket machines at stations that you buy your ticket from, and depending on your route, might be worth the money to get the tram add-on, if you'll be using it regularly.
Also if you're looking for a job, your best bet for part-time jobs in retail/hospitality is looking around your area as a lot of places will advertise in windows. You should write a CV with any extracurriculars you've done that might make you more employable, but everyone has to start somewhere right? I got my first job by emailing a hotel that had (luckily for me) just had a waitress quit the day before. It's mostly luck with that sort of thing. Indeed might have some things, so might Gumtree.