Thing is there are quite a few types of machines if you want to get technical. A typical 'semi-auto' is pump driven and requires you to control the pump. The 'auto' version typically usually some type of flowmeter to measure how much water goes through and stops automatically and most can be programmed. Likely some work on a time based method as well, but I personally have never relied on a machine telling me when an extraction is done as that machine doesn't know what's going on with the puck, etc. Even when we ran a commercial mobile setup with a 2 group machine I still controlled every extraction manually as espresso was the primary focus and deserved the most attention. Then you have the most basic machines like levers, some having boilers and some requiring parts to be heated and water added manually. Then there are spring and manual levers. Not to leave out the classic debated topic of boiler type as in single, heat exchange and double boiler. Of course there are some with no boilers as they rely on thermocoil/block heating systems... it does get technical and does go on and on and on...
For your stated needs I would say look for something like a Gaggia Classic as you can get great espresso and nicely textured milk with one with some practice dialing it all in. If you're OK with buying used Classics can be had at great prices if in decent shape. For the grinding aspect you need something decent. I'd recommend a quality hand grinder if you're OK with that as you can spend say $150 or so on a hand grinder and it will likely be on par with an electric grinder costing in the $500 range. Also lots of used hand grinders floating around to snag. Same can be said for electrics as well. A bit over 3 yrs ago I scored my barely used Silvia/Rocky for $250. It needed quite a bit of cleaning due to the laziness of the original owner as the grinder was full of molded coffee tar, but didn't bother me as that was a steal of a deal with a bit of driving, time cleaning/dialing in and still cranking out fantastic espresso every single day. I no longer have the Rocky grinder, but it's still working great for the guy I sold it to. Quality items will last indefinitely with a bit of cleaning, preventive maintenance and proper use.
After you get the equipment sorted you will need a decent tamper and a scale that weighs in .1 gram increments. Now I'm not anal retentive on weighing every single variable, but weighing your dry coffee doses is rather important for espresso. Don't forget quality/FRESH coffee and balanced water as those are the two most overlooked variables with espresso. It can seem costly, overwhelming, etc., but doesn't have to be. My entire home setup with the Silvia (modified), a Cimbali Jr. grinder, a few high quality hand grinders, the accessories I have for it, the kitchen cart I bought and refurbished as well as my home devised/built roaster I have spent around $1,500 for everything. What I get from this setup is easily on par with the best coffees I've had anytime in the past and I've used LOTS of different coffees for home/commercial use previously coming from equipment costing over $10K. It's not so much about what is spent, but how it's utilized as nothing will trump skill/determination.
Edit... thought I'd add that if you're open to buying used as mentioned you can find something quite usable in your stated price range as people often get frustrated with the time/effort involved with espresso and sell stuff rather cheap. In my area recently there was a deal on OfferUp selling a used Capresso Infinity for $10... yes, TEN dollars and new it'd be close to $100. Not the most ideal for espresso use, but I had one for some time and it worked just fine for my intended purpose.
