Hi and welcome to LL
Are you staying at Beach Club first then looking to go to Kuta? Beach Club is much closer for transport to the Gilis.
For Kuta, Mimpi Manis is a nice clean and cheap homestay, and the owners would be able to tell you where you can go and pay a day rate for a pool. There are also some great local beaches so the kids will enjoy those when you go there. A great way to save money is to stay somewhere like Mimpi Manis and then pay day rates at an expensive hotel to use the pool area.
September is starting to heat up so you also might think about the a/c room or house at Mimpi.
For transport to the Gilis, there are a number of options. Perama operate a boat from Senggigi that goes each morning that is quite reasonable. I forget how much it is, but 'not much'. You could book that while you are at Beach Club. Alteratively you could brave it at Bangsal harbour. You can get a taxi, or local bemo, to Bangsal and walk down to the harbour where public 'ferries' take you to the Gilis. They are not ferries, they are small local boats. Bangsal has a terrible reputation for aggressive hawkers and touts who tell you there are no more boats and that you have to charter, throwing water on your feet and demanding payment for 'washing your feet', telling you the ticket office is closed and you have to buy from them, and all manner of scams. Also for charging a fortune to get you from the terminal to the beach (about a 500m walk I think). When you are dropped off, ignore everyone, and walk to the ticket office which is near where the boats depart on the left. Ferries can stop running around lunchtime, so you need to go in the morning.
If you are staying in Kuta then going to the Gilis you will need to arrange transport back around to Bangsal or Senggigi. Mimpi Manis can assist you with that. For that matter, they can assist you with transport around Kuta including to the spectacular beaches a little out of Kuta like Mawun.
Where-ever you decide to go, while on the mainland of Lombok, make sure you spend some time looking around, go to the waterfalls, the markets, pottery and weaving villages. There is a lot to experience and nothing is very touristy so it is a very authentic experience.
I'm sure you will enjoy your stay.