Not sure exactly what you might want. I have a Ford ZX2 and it's an awesome little car. Has a decent amount of get up and go when you need it and it makes at least 30mpg. The price for most ZX2s anymore is well under $5k.
30mpg?
I get between 45 and fifty...
my first car use to get around 50 - 60mpg. 30mpg
My first car was a '93 citroen AX (1.4 diesel).
since then I've had a '90 nissan micra (1L Petrol) and I've now got a '93 1.8 peugeot 205 (diesel again).
to be honest the car that I've got now checks all the boxes that I need to check.
It's a small hatch back, it's economical, it's fast enough, powerful enough that I've never gotten into trouble, not so powerful that I've ever gotten into trouble. it's got front and rear doors, (so no need to put the seats forward to get kids or passengers into the back).
I'm sure that the ZX2 is a nice car, not proper plush nice, but not old and ratty, for anyone in the UK it's a new shape ford escort.
personally, I don't know if this is your first car, but I generally think that these cars, (because they are slightly larger and slightly more powerful), make good second or third cars. they probably aren't great for first time drivers who are yet to appreciate the proper size and power of some cars...
Car ARE for getting from A to B...
you might say cars are fun, cars should be fun etc...
but honestly, there is very little fun to be had in a cheap car.
really, I've driven other cars and most cars, (especially cheaper cars) are more or less the same...
my advice... take your $5k, and get a $1k car as your first car...
make it a crap run around, make it last, statistically so you're so very likely to write it off that it doesn't really matter what you get. (believe me on that, my first car had more bangs than fireworks night after I'd finished with it).
so get a $1k car. use it for what cars are for, getting from a - b...
also, if you can get a five door, do, there is nothing more annoying than giving friends a ride and having to let them out from the back... (Though to be honest you don't really notice that until you're had a five door car).
and if you really want an exciting drive, use the other $4k that you've saved on the advice you've gotten here to either...
go on a series of track days driving properly nice, and properly fun cars around a proper track.
or, do as I did and get a bike as well as a car.
(I've now got two bikes as well as a car, my Chinese copy dirt bike, and now a Kawasaki ZZR). both of those are far more fun than any car I've ever been in.
and spending wisely, I don't think that I've actually spent anywhere near your five grand budget and for that I've had an old car for 3 years, and a bike for three years and just bought a second bike. (That also includes all maintenance figures, -buying replacement parts etc like new rotors and callipers to pass our road inspection MOT test, though doesn't include garage fee's as I work on my cars myself).
by all means, ignore this advice, get a good car...
Also a bike might not be for you.
but rest assured that if you have a car, for $5k you're not going to get an awesome car.
and if you wrap it around a tree, or another car etc then you'll have no car and no $5k...
so my advice would be look for the cheapest, most reliable car you can find.
drive that for experience. if you crash that it'll be cheap to repair, and if it's written off you won't be so pissed to have lost it.
and you'll still have money left for your original budget to replace or repair the car.