I guess it all depends on how old your children are, but this idea can be adapted to any age group.
Set yourself a budget for the day, say $50 or $100, and see how far it takes you on an adventure around your city. Catch public transport and get a ticket that offers you unlimited travel for the day and just see where it takes you. Do a little bit of internet searching before you go to find out where there are a lot of free activities or get vouchers for discounts, but don’t plan too much. The Entertainment Book (which I believe is available in most corners of the world) is a great source of discounts for places to eat and activities.
Last year I did this with my boys, and they had the best time. We had $100 and the boys had fun budgeting what we could do with it. In Melbourne, we caught the train to Richmond to find a rock-climbing wall I heard about under a bridge. We had to catch a tram, walk through a park and hoped to find it under a freeway. We looked everywhere, but as there was so scaffolding under the main bridge for some construction work, it seemed the rock-climbing wall was hidden, so we missed out. But instead, we found ourselves walking along the Yarra River on a beautiful sunny Melbourne day watching the rowers and cyclists doing their Sunday exercise. We climbed the staircase from the river to Church Street and walked to East Richmond station to catch a train to the city.
We came in at Flinders Street station and walked out to discover the Arts Centre Sunday Market. My then 9 year old and 7 year old boys looked at all the stalls, we had a couple plates of poffertjes, they bought some chocolate, climbed on the sculptures around the Arts Centre and enjoyed the bright colours and eclectic crafts offered by the marketeers.
We then walked along Southbank to the Eureka Tower and purchased tickets to the Skydeck. We saw how amazing our city was from the 88th floor, spotting our amazing sports stadiums, looking at the ships on the bay, and stretching our eyes to see if we could spot something close to home. We grabbed something to eat in the food court, before deciding to catch the first tram on St Kilda Road having no idea where it would take us.
We just happened to catch the No 96 tram to St Kilda Beach. When the boys saw that we were outside Luna Park, they were very excited. We had enough money to have one ride each, and these chose the Scenic Railway.
We then walked down Acland Street, grabbed an ice-cream, walked down the streets towards Elwood and caught the bus home. It was a day for us to be us… a day where we talked about what each of us wanted to do, a day that we all truly enjoyed and were pleasantly exhausted at the end of the day. It’s a memory that we all cherish, with my boys constantly reminding me that they want to do it again, soon.