I am not a lone parent, but for various reasons I try to take my three kids, aged 4yrs, 4yrs and 5 months out of the house for the majority of the weekend.
During the winter we went to soft play, kids cinema and tesco! Now it is sunnier, we spend a lot of time outside at country parks, NAtional Trust places, picnics in local small parks etc. I spend a lot of time cajoling the 4 year olds along to reach some point of interest, while I push the buggy, but it wears them out nicely!
We can manage a whole long day at a National Trust if I take enough food for several picnics. I think the key is to take a lot d things with you and a buggy to transport it all - a picnic (lots, so can have several small picnics whenever the whining ramps up), a ball, a rug, drinks, kids binoculars, umbrella, bag to put 'stuff' in (stones/leaves/feathers, whatever) .... and just keep walking! I take a sling for the baby, and have been known to end up with all the stuff and two four year olds in the buggy while I carry the baby in the sling - good workout for me thrown in to the bargin!
Try not to look at all the simpler families with more adults/fewer kids/older kids/better behaved kids. I used to spend half my time thinking 'well if I only had one child to two adults I could look that perfect too', but I've learned that I have other things instead of easy days out!