It's my hometown.
I agree with @AbzMoz it is worth staying in a few places on the island because the vibe is so different, it's like having several holidays in one place. City + beach (+countryside) for example.
The blazing sun, heat and humidity might be a struggle for a small child especially in Georgetown itself. Sun and mosquito protection is a must, and it will all be sweated off very quickly. General tip from a local perspective - wake up early, do some walking around, scuttle back indoors by mid-morning (lots of air-conditioned malls and cafes), don't go back outside until 4/5pm, stay out until LATE. Sunset and night strolls by the sea is what all the locals do, even tiny children - we don't seem to believe in kids' bedtimes.
Some excellent museums (e.g. Peranakan Mansion, Khoo Kongsi) and "museums" (e.g. Upside-Down Museum, Food Museum) I think we've been to every single one and love them all for their quirkiness - the "museums" are very child-friendly. These are the things we do indoors during the hot hours of the day.
When we stay in Georgetown, we always choose a heritage hotel to enjoy the historical vibe (which sometimes means no pool). The Blue Mansion @TeamGeriatric is fabulous, having said that it is a little bit away from the main conservation area - this is both a bonus (fewer crowds) and a disadvantage (longer walk to the more famous streets) - even if you don't stay there, you can book a guided tour of the Blue Mansion, or go to its restaurant. Some heritage hotels have amazing family rooms which are more like a house of your own e.g. Hotel Penaga Clarke Terrace, or 23 Love Lane.
The beach at Batu Ferringhi isn't as good as it used to be, due to longshore drift. However, if you choose a hotel at the further end of it, the erosion is much less. There are plenty of nice hotels there with gorgeous pools and grounds, too many to mention. But if you have a shortlist I can comment.
A much further beach that is still in good shape is Teluk Bahang. There are a couple of 4/5-star hotels there. But it's a very long drive/bus ride from Georgetown, which is worth bearing in mind if you want to explore both. The advantage of Teluk Bahang is that it is super close (practically walking distance) to Escape - my DC favourite adventure water park ever.
I also concur with recommendations for the Tropical Spice Garden (between Batu Ferringhi and Teluk Bahang) and the butterfly park. Another must-see is Penang Hill, the queue for the funicular is often insane but it's the only way to get up there. There are nice nature walks up there and I particularly love The Habitat, tickets a bit pricey but 100% worth it.