I have had a mixer and mics for the past couple of years but they are not available anymore.

I need mics and a mixer for drums and other instruments.  I do not need professional level equipment.  Just stuff so I can plug in my iPhone and studio headsets and play by myself.  A couple of extra mix inputs for some guitars would be great when I have other people over to play.

What do you recommend?

I would just look on Craigslist, eBay, or Reverb.com for a good used Mackie, Behringer, Peavey, or other name brand mixer. For instance, I see 16 channel Mackie VLZ boards go for the $100-200 range. They may have a dead channel or two, but are otherwise still good. And Behringer? Alot of people mock them because they're made in China, but my drummer has one and it's been a good board. Sounds good and I've seen water drip on it from the ceiling when his kids overflowed the bath tub. Didn't faze it.

For the mics, I would go ahead and get new from an authorized dealer. Too many fakes on the market these days. Hard to go wrong with Shure SM58 for vocals and Shure SM57 for guitar. They're industry standards for a reason.

Get a audio interface and a DAW rather than a mixer.  I have a Scarlet2i4 which I use for bass.  Signal chain either

1- bass -> Scarlet2i4 ->  DAW on Mac -> back to Scarlet2i4 for Headphones/IEM
2- bass -> amp -> mic -> Scarlet2i4 -> DAW -> back to Scarlet2i4 for headphones/IEM.

This is more versatile than a mixer and using the onboard headphone preamp which appears to be what you're looking for.  I can record, mix, create click-tracks with/without voice prompts, and more.  Or use it in a simple fashion like what you're proposing, and even have some patches for amp simulation, etc.

How many mics you use for your kit will depend on the number of inputs you'll need on the interface.  Also, look at the bundled software and make sure the DAW packages bundled are good enough.  Some, like my Scarlet, are neutered too much. If you move up in that line for more ports, you'll get less neutered software.   

Also, I would consider Presonus now if I was purchasing again.  They have their own DAW which is popular called Studio One.  I believe they neuter Studio One less because they don't have to negogiate with a DAW company.

Is this run through a laptop?

Laptop or desktop, either one.

If you go for a mixer, rather than a DAW, the Allen & Heath boards are also good. That or the Mackie would suit your needs.

I actually prefer Allen and Heath boards due to the way they're made, but used Mackie boards are so cheap these days that they're hard to pass up if you're on a budget.

I have an A&H Qu-pac mixer, and it's a pretty cool piece of gear. We've used it for practice, recording live shows and to record our first album. It records to and can play back from a USB drive, up to 16 individual tracks I believe, plus I think a stereo 2 track mix on 17-18, at a fixed 24bit/48k sample rate. Plug in a wifi router and you can control your mix and playback from an iPad (possibly an iPhone as well but it might be a little cramped on the small screen). It also has an LCD display and knobs and buttons for direct controls, and preset/scene saving which was fantastic to go from live settings to studio settings with a button push. I would just plug in a USB stick, hit record and capture a whole jam session. Afterwards plug the stick into my laptop with Cubase, import and mix any keeper tracks. It did lose a mic pre on one channel after I'd had it a year or two (past warranty), but I've read that those are swappable. I just put a 500 series dbx pre in line on that channel and kept trucking. It was just the floor tom, after all. The Qu-pac was kinda spendy at 1500, but there's a newer model, the Qu-SB for 1000 that I would go with today if I was in the market. Still 16 channels but no LCD interface or buttons on the unit. Anyway, if you go the A&H Qu-* route, I can try to help out if you ever have questions. I'm getting a brandy new interface next week since we're focusing more on studio recording and I'm after improved AD/DA conversion and stuff, but I'm keeping the A&H for any live events I have to do.