Beginner Turntable

As far as the best Grado cartridge, it's really up to you to decide because price ranges will vary but if I were to suggest one outstanding Grado cartridge for the money, the Grado Reference Sonata 1 would be the cartridge to aim for. It retails however for $600.00 and my understanding is that this price would currently be out of your range. It doesn't mean however that it won't show up as a second hand cartridge for sale on one of these auction sites.
I'm not real sure about compliance factors as you haven't chosen a table or a phono stage but you'd probably be better off with a high output cartridge.
I suppose if I were looking to buy a cartridge on a budget, I'd pick up the first Grado that I could afford that's being sold as a pre owned high output cartridge but I'd suggest researching on the Grado website before doing anything.
By the way, you'll also need someone to install the cartridge and then you'll have to load the cartridge on to your chosen phono stage. The person selling you the turntable may be able to help you with this.

That said, the difference in your mini system won't be audible or may be subtle at best. Don't overspend if you are going to keep your mini system.

Meaning that since you have a budget sound system, it would not make sense to buy an expensive high-end TT and cart. You would not be able to fully appreciate the high quality sound and the benefits that come with it...such as more detail, better dynamics, less surface noise on the LP.
If you were to upgrade from the Denon mini to a more revealing system, then you could easily hear the difference in sound quality between TTs and carts.

So if you are planning to upgrade from the mini in the future, then you should buy the best TT you can afford.

Hi Lowrider, I have to agree with you that a better table than a USB or entry level modern table might be advantageous in the long run. A Linn or Thorens would provide the chance to make upgrades (i.e. a solid plinth and a better tonearm) along the way without breaking the bank up front for the initial purchase.
Also, turntables are the one piece of equipment that I'd have less hesitation about buying used because they're more mechanical than electrical.

I don't know how many of you have heard a Pro-Ject Debut Carbon, but it sets a
new standard for what you get for $399. Add a phono stage and it's right in the
OP's budget w/no fiddling, upgrading, refurbishing, or whatever. You can even
upgrade w/o mounting a new cartridge just by replacing the stylus with the one
for the 2M Blue.

It's so good for the price that the next logical upgrade would be the downstream
components.

Exactly what I meant, thanks you Lowrider57.