Less than ideal router strength, curious what my Tech Tilderinos would recommend

First things first - is your current Verizon-rented router 2.4-ghz only, or both 2.4 and 5ghz? Your laptop or phone should tell you this somewhere, or the router should have markings for 802.11ac if it supports 5ghz. If it doesn't support 5ghz / 802.11ac, then it's outdated and you should absolutely upgrade.

If you run Ethernet across an apartment, either of Cat 5e or 6 should work. Cat 6a, 7, and above are wastes of money for your purposes. Even Cat 6 is a bit overkill for you but the price difference between 5e and 6 is typically a rounding error so buy either one. Monoprice is a great place to buy inexpensive but well-built cables.

Switching from a rented router to one you own is absolutely a good idea, it'll usually pay for itself within a year. I run Ubiquiti gear personally, which I love but requires some fairly complicated setup that it sounds like you might not be comfortable with. For less expensive networking gear, I've had nothing but good experiences with TP-Link - I have several of their Ethernet switches and "smart" power outlets. Here's an inexpensive but well-reviewed router & AP, for example.

What is your upstream connection to Verizon (Fios, cable, etc?). The device you rent might be an all-in-one unit that combines modem, router, and wireless AP - if you replace it with your own you'll need to replace all the functionality it provides, possibly using multiple components.

Things like wi-fi extenders and powerline Ethernet are all options, but I'd only pursue them after upgrading your core router, and only if you're still unhappy with the performance.

I don't know if you've considered this, but another option could be powerline ethernet adapters, although they may require a bit of configuration in an apartment building. This would include preventing other people from connecting to your powerline adapters, and making sure that both power outlets are on the same circuit.

EDIT:

There are also adapters for coaxial ports, if your apartment has those.

I had this exact same issue when I lived in an apartment. Lots of people are recommending better wireless access points, which is fine, but it doesn't address one major issue with apartment networking: radio interference. Consumer wireless networking devices only have a limited slice of bandwidth to work with. In a crowded place like an apartment complex, you can have hundreds of devices in range of eachother transmitting on the same small selection of frequencies. This can drastically reduce the range and reliability of your wifi. While it will probably still be fine for web browsing or even watching video, it can manifest as big lag spikes when playing online games. 5GHz wifi mitigates this to an extent (there are more bands to work with), but has it's own problems with not penetrating building materials as well.

I ended up buying a spool of Cat 5e at Home Depot and running it under the baseboard all the way to a gigabit switch I placed behind my entertainment center. You don't need the stuff with a plenum sleeve. It's legally required if the cable is going in a wall or ceiling, but for this purpose it just costs more money while making the cable bulkier and less flexible. A higher grade cable like Cat 6 will also have the same problems, and unless your apartment is very big it shouldn't be necessary to achieve a gigabit link.

Repeaters are really cheap and easy to set up nowadays. $20-30 per repeater, all you really need is a power socket. You can spend more for fancier/better repeaters, but honestly multiple cheap ones is often the best solution as you can maintain a strong signal.

Alternatively you could do PoE (power over ethernet) where your house's power lines are used as a way to transmit eternet signals. These tend to max out around 300mbps, however and at this point I wouldn't recommend them over a wireless repeater unless absolutely necessary.

Have you looked into mesh network routers?

If you're unfamiliar with those, they're basically repeaters but with better software and hardware integration. You put a base station with your modem and then spread the satellite units throughout the house to spread the coverage out and fill in the problem areas. The satellites communicate directly with the base station via a dedicated wireless channel.

There's tons of them out there from different makers and I'm sure they all have pros and cons, but I can only vouch for Netgear's Orbi routers. I've got three in my 1,500 square foot house (one base station and two satellites). They work great and I've never had any issues with them. I have one in my TV room, with all of the game consoles plugged in to that via ethernet, which then talks to the main hub.

Run a really long Ethernet cord - possible but would be a little bit of a project. I looked into it a little and I'm curious if there is any appreciable difference between Cat 5, 6, 7, or 8 from a home user perspective. Also, what's a good resource for buying one, since Amazon is not really trustworthy anymore?impossible

I can't see anyone else suggesting a Powerline. My router is downstairs on the opposite side of the house, it would be pretty difficult and destructive to route a cable that distance.

I bought a cheap (40£) powerline kit, and got one with passthrough. The router's connected to the socket downstairs which is on a power strip, and the one upstairs is connected to the main PC, which I turn on and off as needed. It's working fine and the data rates are slightly higher than I would get via wifi.

EDIT: I see now that someone recommended PoE which I think is the same? Hm.

EDIT2: Oh, I really should have grepped for it instead of looking manually, ha.