plant some inexpensive ground cover between the patio stones and throw grass seed over the lawn.

I agree with the other comment on clean up. Sometimes the job seems overwhelming. But if you start by putting the shovel away, the toys, and all the other stuff you start with a clean deck.
You have a nice basement window, it would look prettier if everything was trimmed around it.

You can then sweep up all the dirt off of the steps, and patio.
When you get to the bottom and all is clean, then the fun can begin.
Wood stain on the exterior wood perhaps? Pick 2 or 3 paint colors and do a pattern on the concrete patio part. Nice little table and chairs, umbrella. Nice spot for lunch with the kids, or coffee in the morning. I started many years ago not knowing how to do
anything, and not having much, and I just kept trying to make things nicer. You can do it,
don't be discouraged.

You need to talk to your landlord. The wood needs to be replaced and re leveled it is rotting and your little ones will get hurt and splintered playing there. Not to mention rotted wood attracts termites.

There are safe weed killers that will not harm the children or pets but will kill unwanted grass and weeds. One way is to boil water and toss it on the stones. Another is to smother them by placing newspaper over them and a rock to hold the paper so sun no water...dead. Mix vinegar and salt that will kill slugs if you have those too. Once the weeds are dead they have a habit of coming back keep a spray bottle of vinegar to zap anything green that is not where it's supposed to be.

Check Craigslist many people are happy to get rid of a swing set for free.

Those stairs are a major safety hazard what with the pavers being lower than the wood. What if you have to run out of the house for your child or if there is a fire? Call the city permit department so they can contact your landlord. Those stairs are totally unacceptable.

And by safety hazard I mean tripping hazard.

Lucidos, thanks for the weed killing tips; I didn't know about the newspaper...will use that in my own yard.