All the advice above is excellent, and I'd encourage you to remember the 'armour' quality of good clothing too. There is a ton of evidence around how both (1) people respond to visual cues/presence and (2) how much clothes can give you confidence. You said it yourself, this is your first formal work experience so you want to feel and look quite different from how you did, say, in school uniform. That's the issue that
@Jessica Booker has picked up on with the shoes IMO - they are just a bit 'rough and ready' for the office. Fine as a statement, and very practical, but just not very elegant or professional. The simple, understated slingback or low-heeled court shoe she linked are just going to help you feel that much more put together. As long as they are the right size...!
Regarding the bag, it's less about 'looking stupid' - so many people carry backpacks these days with their laptops that it isn't so crucial. But you are quite right about wanting to feel a bit more relaxed. You want to be able to go for a drink and not have to keep the backpack on, while being able to buy a round if that's the right thing to do. You could consider using the backpack for your work shoes, umbrella etc, and having a separate smaller bag for your wallet/whatever.
Mango can have fun handbags (Spanish so tend to be stylish), also maybe
Zara; M&S also have some good designs (this looks a slightly pricey but very well-designed
camera bag , also in
faux leather for about half the price), but it depends on what you feel like. Bags are something that you need to play with to find what suits you. You mentioned Uniqlo and they have a couple that look OK - larger ones you could compartmentalize with zip bags like pencilcases etc? I prefer structured bags as I hate not being able to find anything, so that would be my workaround. I also try to have everything zipped away; pickpockets have had a rough old Pandemic and are out on the prowl again, not a trauma you need on a VS. So look for something you can open and close easily. EBay is brilliant if you know what you want (eg
M&S saddle bag or
Zara) but then you are rather stuck with it.
By the way, I don't own an M&S handbag but it's a good place to start. Accessorize always used to have some hidden gems so you could search for 'Accessorize suede bag' on eBay or something? Far more sustainable too, but check the listing to make sure it isn't going to smell of parties and cigarettes (!).
Another way to go would be to ditch the student backpack and get a
tote bag - but you'd still need something for your essentials.
In the end, they aren't going to hire or not hire you based on your handbag but you might find it just gives you that chance to feel a bit more professional. Have fun!