That sounds like quite a large area to mow with a push mower!
The cylinder mower (which 'push' is) can give the best finish of cut.
Electric rotary mowers are OK though, but the Flymo 'floating' sort don't really float quite as lightly as the adverts imply.
So a wheeled rotary is probably easier, particularly if the grass is long, and there are slopes and the damp area. We have a Wolf wheeled one; all Wolf tools certainly are not the cheapest, but they are versatile and reliable, and I have used a lot of them.
Whatever sort you get, make sure it has a container to collect the cuttings; some 'compact' the cuttings into a smaller volume, but I don't know how well that works. Also a range of cutting heights that are easy to adjust is useful, at least three I would suggest.
A manual lawn rake is useful to rake up debris, and I used to rake BEFORE mowing to stand the grass up. Wolf do a wheeled 'scarifier' (which has VERY sharp points on its blades) which is useful once or twice a year to get 'thatch' out of a lawn.
If you have a mains electric mower it MUST be protected at the wall socket with a RCD safety cut-out, in case the cable gets cut, protects against electric shock!
Dandelions: a lawn 'weed & feed' fertiliser will get rid of them eventually, but it may take some time. There are 'gadgets' you can screw into the ground to pull them out, but that is quite had work, and needs a 'T' handle, and Wolf do those.
Soggy area: spiking with a large fork may help, or you can get 'hollow tine' aerators to take out plugs of soil. Then brush in peat or sand might help.
SAFETY: with a toddler, make sure he cannot get at tools, electricity, or chemicals.
Hope you make some progress!