Worst mistake I ever made was teaching myself. Which makes buying a set just to get matching clubs the second worst mistake.
Find a PGA teacher and start with the fundamentals. That person can then recommend which clubs to start to replace to help your game at that time. Getting fit for a whole bag while just starting might require upgrading within a year. None of us have the same mix of clubs as the next guy. Because we are all different.
Enjoy getting into the game. Not sure where you are located, but there has been an explosion of indoor simulators in WI to enjoy from Thanksgiving to Easter.

Yes, probably not worth trying to spend money altering your existing set. If your budget allows, do a new club fitting and it will at least get you started into a set that generally fits your basics for length, loft, lie angles. You can use that to purchase new or find some good used set that closely matches your fitting results. Over time, as you get better, you will want to dial in your next fitting for your improved swing which will likely alter shaft flex and clubhead type. I usually recommend a new golfer split their initial budget between equipment and proper instruction. Getting good fundamentals from a professional will greatly accelerate your development and enjoyment of the game. Have fun and great to hear you have chosen to pursue such a fine sport.

Everyone here is going to tell you to get fit, but that's not smart money. Get a good quality set of used standard clubs and start working on your game, hopefully with some good instruction. Once you feel comfortable and develop a consistent workable swing, then you can consider getting fit for clubs. If it doesn't work out, you're not out a ton of money, and if it does work out you can always sell or trade the used set when ordering a fit set.
And this is actually a good time to start. Find some indoor lessons and start working on it, and order a fit set of clubs by February, and next spring you'll be set to attack the course.
Good luck and have some fun.

This is the correct answer. You are unlikely to have any real consistency in your swing, making fitting a total crapshoot. Keep the clubs you have for now. Get some lessons- or even just read a basic golf instruction book or utilize online instruction content- there's plenty of it to get you started. Grip. Stance. Swing sequence. Then seek instruction to hone those basics. Once you have a swing that produces a somewhat consistent result AND you want to invest in equipment (you don't have to have new, fitted clubs to enjoy golf) to improve, then go get fit.

Your story mirror my early days when I took up the game. My first "set" was a mismatch 3 - PW where each club was either from a different manufacturer or the same manufacturer but a different model. The shop where I purchased them from, marketed them as a "starter set." After a few months, I was bit by the golfing bug and bought my first real set: Titleist DTR.

Today, I would approach things differently as there are many more options for a beginner golfer. Bring your set to a well stocked pro shop that offers demo clubs for you to hit. I would compare those clubs with your existing ones and see how they perform and feel. If you see a positive difference then you may consider getting fit. If not, stick with your existing set and get some lessons to see if improvement can be made with your swing.