What cycling accessories are actually worth spending extra money on?

$15 is dirt cheap for a new saddle (seat). What else? An under-the-seat bag, a spare tube, tire levers, fold-up tool, and either a CO2 cartridge or a frame pump. If you plan on hauling anything, definitely a rear rack and panniers and/or a rack bag. Insulated rack bags are good to take along spare water bottles & keep them cold.

If anything at at all get a cycle computer. See how far you went on a ride and fast you went and what your average speed was. See how fast you are going at the time and how much time you spent on that ride. So things crap out every now and then, replace them as you go along. After a while you'll be wanting new bike by then you'll know what you want and how much time you're spending on the bike.

Cycling shorts aren't cheap but if you can get a pair for every day of the week, same with the jerseys. I too can go two weeks with having to do laundry.

Upgrading when components wear out is a good idea. Upgrading because you bought a cheaper bike than you wanted in the first place isn't. People who buy bikes thinking they can turn that bike into something better are wasting their money.

I own enough cycling shorts and jerseys to go almost 2 weeks without doing a washing

Get things that you can use on the next bike. Water bottles, frame pump, compact tool kit with tire levers and patch kit, spare tube and under saddle bag to keep it is a must have Cycling shorts and jerseys are are nice 2 pairs of shorts and 2 jerseys will do. Pedals with toe clips or cycling shoes with clipless pedals will help. Cyclocomputer and heart rate monitor are nice. Helmet, sunglasses with a mirror will keep you safer.

Get clothing at Performance Bike or Amazon.

Upgrade your cables if the bike is more than a few years old. Get some quality cables such as Jagwire or Gore. The shifting difference is awesome. Shifters smooth as butter!

BA:

I have at least a dozen jerseys. They just accumulate over the years and jerseys seem to never wear out. I use 3 pair of bib shorts per season. I do my laundry often or on multiple day ride events I wash the shorts in the shower as they hang dry quickly. Don't let the funk in your clothes begin. It seems hard to get out. Wash your hands after a hot summer ride while they are still in your gloves thus cleaning the gloves of sweat. Gloves can really stink so stay after them. Never wear a pair of socks more than one ride before washing them. Wash your head band after each ride and your head pads in your helmet weekly in the hot months. I ride every day it's not storming.

Helmet

Head light if you ride in the dark

Tail light if you ride in the dark

Pump

Patch kit

Tire levers

Extra tube

Multi-tool for bikes

Seat bag (or other bag), to hold the patch kit/levers/tube/tools

Water bottles

Bottle cage(s)

Fenders if you ride in the rain and your frame can handle them

Rack if you like to carry stuff and your frame can handle it

Panniers if you have a rack