I've never changed my strings before by myself, and figured I may as well learn to do it soon. I have a Fender Strat.
If I change the strings by myself, can I mess up the action of my guitar? Like, I don't fancy messing with the bridge etc to alter it. Can stringing a guitar badly affect the way it plays/sounds? Thanks!
Yes, if not done right changing strings can lead to problems, to a certain extent. If you don't get the strings set in the bridge saddle or nut slots right it'll cause obvious problems. Also getting the right amount of winds around the tuning pegs is key.

They won't throw off the intonation(bridge) or anything else unless you're changing string gauges, and even then probably not by much.

I would recommend this video for help on how to change strings:

Epiphone Les Paul Standard Pearl & Ebony • Les Paul PlusTop Pro Honeyburst • AJ220VS • Squier Standard Stratocaster CAR
Marshall Class 5 Combo • Digitech HT-2 • Vox V847 • MXR M68 Uni-Vibe • Soul Food • BOSS SD-1 • Digitech RV-7
As long as you're changing the strings correctly, you're not changing gauges, and you do them one at a time (as opposed to taking them all off before replacing them) you have absolutely nothing to worry about.
Like doing anything for the first time, just go carefully. Its really doubtful you would mess up the action or alter the bridge. I screwed up the amount of windings(cut the string short) around one of the tuning pegs my first time and had to buy another set of strings.
When you take a string off, check the that the nut holding the tuner is still tight, sometimes they can loosen up.
You'll do fine. It gets easier each time.