Oh gosh, I had a similar age gap and I genuinely can't remember much of that whole first year which I lived in a haze of sleep deprivation (2nd DC wasn't a fan of sleeping at all!). I didn't cope very well (PND) and felt like a zombie much of the time. We had just done a long-distance move, were away from family and had no friends locally.
If you have family and friends nearby this is definitely the time to use them, take any offer of help you get, and sleep any second you can (it is nothing like the same as a good nights sleep I know). I sent my 1st DC to a local nursery, earlier than planned, at 2.5 for a few mornings a week just to have a little quiet time where I could slow down and not deal with toddler energy levels (and very occasionally nap). I used the tiny bit of time in the evening / when the eldest was at nursery to prepare a couple of activities for them to keep them busy...I'd set up a tray with a craft activity, dig out a book from a pile I kept hidden away and set up some toys in a corner ready. It didn't mean I could rest but at least meant I didn't have to put too much thought into entertaining them when I was tired. When I did cook I'd always make double meals and freeze so it gave me at least 3/4 days a week when I didn't need to sort dinner.
Oh and TV. This is a time where any rules on tv can be stretched. If you can grab 20 minutes to sit down with your eldest while they watch tv and you have a cup of tea, then do it.
The main way I coped though was by having a lie in at least 3-4 times a week. So after baby was awake and fed and toddler was awake too DH would take them downstairs and I'd go to sleep for 2 hours. It's not much, but it did mean I was vaguely more coherent.
Sorry, I know that's not helpful in actually solving the lack of night time sleep (the truth is we never did solve it, despite trying everything), but I hope there's something useful in there.