I had the same budget when I went to buy an espresso machine so I'll summarize what my research revealed.
If you don't have a good grinder and don't mind a bit of work you can pick up a good hand grinder for $40-$80 for espresso....then you can spend most of your budget on a machine. Cheap electric grinders just make a mess and are a waste of money in my experience. I don't think I would want anything less than a Baratza Virtuoso Preciso if I was going to buy an electric...but that would eat up your whole budget.
There's a bunch of cheap espresso machines below $300 but almost all have pressurized portafilters which "fake" crema and cover up mistakes...they'd be fairly easy to use but won't make great espresso. You can use ESE pods with these and avoid buying a grinder...but ESE pods are like coffee teabags...they're convenient but are full of stale coffee and you'll never get a great cup from them.
You could go for one of the low end Gaggia machines they have a bunch of models which aren't very different which is a bit confusing...but Gaggia uses standard portafilters on their low end machines so you should be able to make real espresso with one.
Problem with any new semi-automatic in this range is build quality...make sure the vendor you buy from has a solid return policy in case you get a lemon.
There are some other options like the mypressi twist which is designed as a travel espresso machine...its tiny and looks like a weird portafilter....but it makes real espresso and is inexpensive. It doesn't heat it's own water, you'll need a kettle. And it doesn't steam milk but you can use a stovetop milk steamer. A hand grinder, plus mypressi twist plus a stovetop steamer would be a good if unconventional setup that should fit comfortably below $300. But pressurized gas cartridges will add up if you make a lot of coffee.
You might want a capsule system like a K-cup or Nespresso...those machines aren't expensive but the capsules can add up. But it's easy, can make ok coffee and you won't have to worry about getting a grinder.
Thinking about it...you could go for a moka pot plus a grinder and make coffee like most Italians do in their homes

It's not quite espresso though...the pressures are too low.
Or you could go the route I did with a $300 budget...I hunted ebay for a used manual lever machine and bought a new hand grinder. Problem with this is it's probably the least convenient way of making espresso, it's very hands on. And you need to be careful you get a machine in good condition otherwise you could dump a lot of money into repairs if something like say a heating element needs replaced. But this has the potential to get the best results in the price range if you're willing to work for it.
so...a lot of tradeoffs no matter what you do...but there are a few options.
