Firstly really look at your budget to make sure you're getting the best price for all your essentials and that you need/use all the things you pay for.
A few little savings can add up to a noticeable improvement in your disposable income, so you can spend a bit more freely on the things that are fun but not essential.
Also look for little ways to make extra money. I have a spare bank account with 2 direct debits coming out of it that I switch regularly for the introductory offers - in the last 6 months, I've switched it to Lloyds and now Natwest and picked up £250 cash and 6 free cinema tickets on the way, for probably less than a hour's admin. Get the Moneysaving Expert weekly email to find out when these are available, and also other offers and discounts.
Evaluate the cost of everything to make sure that you've got the best price and it's worth the money. Don't sign up to endless subscriptions for streaming services etc. You probably can't watch more than one or two at a time, so cycle round if there's things you want to watch on different platforms. Plus Now TV always have offers so it's rare you have to pay full price. I'm currently on £2.99 for the entertainment package for 6 months.
Eating out and takeaways (plus coffee and bought lunches) are getting more and more expensive and add up to a huge spend if you do it regularly. If you're getting takeaways because you don't want to cook, keep nice supermarket pizzas, curries or whatever in the freezer instead, or look for supermarket meal deals. Costs about a third of similar food from a takeaway and with restaurants, the saving is often even more, because you're also paying high prices for drinks.
For days out etc, always look for offers, big attractions often do free child places or similar, its rare that you have to pay the full price for everyone. Or if there's somewhere nearby that you could go regularly, buy annual passes (I believe that sometimes you can get deals on these at certain times of year eg, Black Friday, the beginning of the summer is probably not a good time to buy these for theme parks or similar). Many Mumsnetters suggest National Trust membership as a good way of regular cheap days out, it's also often useful for free parking on certain beaches.
Of course, take a picnic rather than eating in the cafe at the attraction, or perhaps just have an ice cream (if you really want to save on ice creams, buy a multipack in a supermarket or convenience store instead, but that doesn't work if you want it while you are actually at the attraction). In many places the food is expensive, poor quality and at the end of a long queue so it's no great hardship to avoid it anyway.
Some great ideas there too @SweetSakura . If you can go for walks exploring your local area, that's free and obviously better than going to the shops 'for something to do and get out of the house' if you end up paying for parking, buying coffee/cake or fast food and also things you don't really need.
Look out for free things to do in your local area or not too far away. Galleries and arts spaces are often free and could have craft activities for children.
My city has loads of 'Little Free Libraries' which are mini libraries that you can just go and choose a book to borrow, or donate some of your old books.