Maybe start with a dog run so that that at least will help. I have huge dogs they have a doggie door to the dog run it is done in gravel for the most part with a deck and a small patch of ground cover in one area. The gravel makes cleaning up the poop easy and the ground cover surprisingly thay rarely poop on but they like to lay on it and they all love the deck.it is under an overhang from a deck above so nice shade and water proof . Then post a to scale plan of the rest of the space so we get some feel for the size you have. Also what do you want from this area like seating, fire pit a bit more info is always good.I think a multi level deck could work if you took it out from where you have the deck now instead of all those stairs so a deck as high as the bottom of the windows on the left and out into the yard quite a ways so that you have a nice outdoor kitchen and eating space on the top level and then nice seating on the lower one you might even be able to go big enough to have a propane fire pit too

Some ideas

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I’ve posted pix of our patio and adjacent gardens to help others, apologies if it is redundant. Trees provide nice shade and our seating will easily accommodate small or larger groups.

I'd start by painting out the red and planning a first story patio area

maybe do a composite deck on the ground level

bring in some hardscape, maybe a retaining wall design that can double as a bench

firepit

hardscape ideas for a more modern look

a lot depends on budget. why not hire a landscape designer?

These are all great ideas so far! We are working with a smaller budget and will likely need to phase out some of the work. Any thoughts on if we should consider extending the deck this summer and then next summer concrete under the deck or vice versa? Or just leave deck as is and just have a large patio? Pavers are out of the budget, it would be regular concrete (no stamping).

You can use gravel for making ground patio.

Maybe have some patience, come up with an overall plan that meets your living and use patterns, then save some money for that plan. Outdoor living should be designed with the same fervor as kitchens on Houzz are scrutinized. Like how big you want the deck and is that your primary use; whether the basement connection is primary or secondary; whether the firepit is on the deck (so propane table) vs ground (maybe masonry gas-assist); whether you can achieve the privacy on a 2nd floor deck you want; how the stair placement doesn't interfere with seating space and may circulate to other areas of the yard; how the shape of a ground patio relates to softscaping; etc just some examples. As well consideration that your deck may be inadequate (e.g. ledger board on cantilevers is not to code).

Here are some ideas:

Are you handy and willing to install a patio DIY? I designed then installed our dog run, paver patio, egress deck, pergola, firepit and much of the landscaping. It was alot of labor, took longer using mostly evenings and weekends but reduced cost by over half.

Building upon comments from @3onthetree, create a plan. Get books about patio design and construction. Measure and stake it out. Do it!

Here’s a drone perspective of our lawn