Specifically, what is the avenue one takes to get actual control over the buying and selling of stocks with their own money?
There are a growing number of "free" stock trading services and apps, and even "traditional" brokers offer individual stock trades (note that you can't usually trade individual stocks in a 401(k), but that's a different story).
A "day trader" is simply one who buys and sells the same stock in one day. The pattern is to get in to a stock, hope it goes up, and get out before the day is over. You can trade individual stocks and hold them for long periods of time and not be a "day trader".
So the "avenue" is to pick an app or broker, or if you already have an broker for your mutual funds, use their stock trading services. You can choose whether you want to start a normal investment account (where gains are subject to taxes), or a "retirement" account like an IRA, where the tax is deferred until you withdraw the funds, but there are limits on what you can contribute and withdraw.
Depending on the broker or service you'll have to deposit the cash to use to buy stocks. Avoid margin trading like the trade unless you are absolutely certain you know what you are doing. With margin you can lose multiple times what you start out with.
Do note that individual stocks can pose incredible risks. The last few days have taught many people that stock price can become completely disconnected from the value oft he underlying company, and if you're on the wrong side you can suffer massive losses. Even "informed" investors can get blindsided for completely unexpected reasons. That's one of the big benefits of mutual funds (or ETFs if you want more control of the price you pay during the day), is that they spread this risk over hundreds of stocks, so even if one "stonk" goes bananas, your exposure is limited.