This.
It is almost never 'not worth saving' because you need to have money available for annual and irregular expenses like replacing white goods, paying for other big purchases, insurances, Christmas, holidays etc etc.
On the matter of declaring the interest, each account will issue a statement at the end of the financial year. Keep all these and just check that you don't earn over £1000 in interest (spoiler: you almost certainly won't in the short term, but keep reading anyway) and you probably won't need to do anything more.
However, up to now, very few people will have earned more than their interest allowance due to low interest rates (if you've had 1% pa interest, you'll have needed to have £100k in savings to beat it, so this only applies to people with a lot of savings, or high earners who usually need to do a tax return anyway, which is how you declare it).
If interest rates keep increasing, the amount you need in savings to exceed the £1000 interest allowance will go down quite quickly - eg if normal accounts start paying 5%, you'll get £1000 interest if you have £20k in savings, which many more people will, especially if they are saving for a house deposit - but this will be overcome by people using their cash ISA allowance again, which has largely become redundant over the last few years due to poorer rates on ISAs and the £1000 allowance.
For teaching yourself about personal finance, Moneysaving Expert is an excellent resource. There's articles about just about anything you could imagine (savings, pensions, debt, budgeting, cutting expenses, coping as best you can with the cost of living crisis, house buying, etc etc) and a weekly email which gives ongoing tips and helps you learn about money in small weekly chunks.
Also Meaningful Money, which as well as the information on the website, has a podcast, youtube videos and a book. This is more about investments, but also covers the basics of budgeting and preparing for retirement, which is a lifelong journey.
Finally, I must ask, is there a reason you're working PT? If it's to bring up DC, you should be aware that this puts you in a vulnerable position if you're not married, although you do seem to be saving some extra towards a pension, which is better than many women who have limited their earnings compared to the father of their DC has done. Something else to think about.