You’re near the old trade fair district and not far from City Life which is a new area with shops and restaurants and activities, within the now hugely expensive real estate area with the designer skyscrapers. There seems to be a free canoe activity for children which might be fun. Bear in mind that Italian schools break up at the beginning of June so there are a fair few activities available, many free or low cost, for the children who are not sent away to their grandparents at the beach. Ask at your hotel if there is a list of activities.
For restaurants, that’s an area I know less of as it has changed a lot since DP lived there. I will ask around but it might be easier to ask at the hotel. Sorry! I’m not being of much help! I do know a restaurant which serves food from the Valtellina (area near Milan in the Alps) but it’s quite stodgy stuff for mountaineers and skiing all day so probably not ideal for a hot day. It is very good though.
The swimming pool is in Porta Romana, you can get a number 16 tram there from near Duomo or the metro is 3 stops on the yellow line from Duomo station (exit Viale Montenero side). You will need to walk a few minutes when you get off. Beware! The water is fed into the pool from an underground spring and is very cold. The spa is nearby if your mum fancies that while you’re in the pool.
For public transport get the Moovit app, it does sometimes go a bit doolally but I’ve found it invaluable when I’ve needed to get around on public transport in foreign cities. Not least because they’re building a new metro line in Milan and a lot of bus/tram routes have been changed because of the building sites. It will help you to identify where the stops have been moved to.
Re clothing, yes to floaty, yes to natural fabrics. Make sure shoulders are covered for the Duomo tour and wear comfy shoes as there are a fair few cobbles in the centre. Birkenstocks are perfect, also worn by lots of Italians so you won’t stand out. What does separate the foreigners is flip flops in a city. Total no no for the locals. But you do what you feel comfortable with.
I hope you have a great time, there are lots of lovely hidden corners to Milan and many are lovely. You’re not tripping over ruins or renaissance architecture like in Rome or Florence, but Milan was quite heavily bombed in WW2, unlike many other Italian cities (the Galleria was bombed and has been totally rebuilt), but it is imo very vibrant and more cosmopolitan than some other Italian cities. Much as I love Venice and Rome, I’d much rather live in Milan.