I thought I'd start this as a general things to do, as opposed to actual events (which I am going to add to later, but anyway)
So Please add your best things to do or places to go free or otherwise.
Brynmill park here Lovely park with new playground(not on map yet), walk round fenced lake, toilets, small kiosk shop that sells duck food, hot drinks and snacks. Ducks to feed but no squirrels, lots of dogs around. Discovery centre is open in the afternoons and has jigsaws, colouring and activities inside, but no eating or drinking inside so no good for rainy picnics. Picnic benches and lots of mostly dog poo free grass (very clean) Good for toddlers and bikes/trikes/scooters as it is nearly all pathed except for one steep hill. Can get very busy in good weather.
Really nice quiet park, lot of dylan thomas stuff around here as well, he was born down the road and there is a shelter and stone dedicated to him. Very hilly park but great playground with low toddler climbing frame as well as larger one for bigger children. Very clean. Lots of squirrels and some dogs. Free (I think) Tennis courts. Good for slightly older children as some of the paths are quite steep but lots of good climbing trees and benches as well.
Waterfront museum (the new one) here Free entry, nice cafe, quite cheap with lunchboxes/hot choc and a play area with lego and books. Not a huge amount to see that small ones can understand but lots of 'interactive' exhibits means they get to push buttons and use roller/scrolling balls which keeps mine quiet even if they have no clue what the voice then talks about. Drawers that open and energy 'machines' in each room help. Small play area upstairs at the back with bricks and colouring means you can have a nose as well. mostly local history with videos and boating/sailing/harbour stuff with a good view over the marina phase 2 as well. Lots of friendly staff who don't seem to mind screaming children running about having fun either (which helps)
Swansea Museum (the older building across the grass from the waterfronthere Really cool tiny little museum, also free. If you ask at the desk when you go in they will give you a free 'Mouse hunt' board which is a sheet of A4 with pictures of various types of mouse on (fishermouse, sewingmouse, druidmouse etc) a clipboard and a pencil. All around the museum are small plaques on the skirting with matching pictures and they have to find them all. Suitable from 3ish and keeps them nice and busy while you look round. It is very cramped and not suitable for buggies but if you ask nicely they will let you dump your bags/coats/buggies in the private library bit while you go round. Exhibits downstairs change in 2 rooms and they have a small childrens table in all the rooms with an activity, either colouring or a light box or fuzzy felt etc. Upstairs is an egyptian mummy, the roman room and the 'room of curiosites' which is the bizarrest mix of crap you have ever seen. It has a stuffed condor, butterflies, a victorian parlour and a welsh kitchen, amongst other stuff, it's very cool. No cafe, loos are right in the far corner downstairs, adn the stairs are very steep but there is a very old dodgy lift. I love it though.
We (or rather my boys) love Blackpill Lido whilst I love the Junction Cafe. It is very handy to be able to park across the road, have a little stroll along the seafront and then have a paddle in the Lido.
Plantasia Under 3's free, adults £3.90, kids £2.95 You can get season tickets as well.
Good for rainy bored days, lots of plants (yawn) but also in the foyer are bugs, piranhas, cockrocahes <shudder> etc. There is a big playroom with a 'DO NOT TOUCH' exhibit, really unhelpful and tables/activites. Inside there is a butterfly room with parakeets so always soemthing to see when butterflies are dead out of season. Big snake and a small family of monkeys at the back are sweet. best are the Koi carp, masses of giant feckers who you can feed if you have remembered 20p's not good for fearless running climbing toddlers (lots of wet, possibly slippery walkways and climbable fence round pond) but okay for preschoolers and buggies. expensive gift shop and cafe, but you can buy Joe's icecream and donuts from the shops right outside. Just becareful, it's right next to toys r us so make sure you go park near mothercare unless you like being whined at for toys.
Oh yes, Lido is lush and free! (apart from £2 for car park) but the Junction cafe is overpriced imo(am cheap though) I like the playground next to it as well, and in the summer they have a bouncy castle/carousel stuff up as well.
TOP TIP if you go veyr early the water is colder but it is dead, most people arrive after half 10ish and it is packed by midday ona nice day.
More about Lido Park in Clyne gardens car park pay and display, you have to cross oystermouth to get to the lido but then it is a little path down so safe for littlies to walk. The lido itself is normally very clean and has fountains/sprays and a rubberised floor so it isn't too slippy. it doesn't get deeper than about a foot. There is also a 'big kids' playground next to the lido, sandpit, big seesaw and slide/swing things and lots of grass for sitting/picnics. The other side has a toddler playground with little toys/benches which is sweet. If you get bored of that the beach is lovely at this bit and there are lots of shells, no rockpools round here though.
The swansea bay rider leaves from outside the Junction cafe just next to the lido itself and there is an icecream stall outside if you don't have more money than sense. Public loos behind the icecream place are dingy and not veyr clean, but still exist, which helps...
Leisure centre. <disclaimer - as I have 2 children I cannot go into the swimming pool with them both and haven't been for ages, so will not comment on that!> I love the Play zone in the LC. Mine are 2 and 4 and we have been going since the small one was a few months old. it is very clean, and very open and bright, so you sit in actual daylight while they run around in a mesh covered cube. This means that you can usually see them the whole time they are inside, veyr handy. They have a baby area for under 3's with light display/bubble thing and ball pit. I really like it, especially as you can take your own food and drink without it being a problem, and you can use Tesco Clubcard vouchers to pay. And they have working wifi, a coffee shop opposite which sells very good value lunchboxes and also has it's own small play area. The only pita is you can only pay for an hour at a time.