I am somewhat versed on points and miles strategies but I know there are others here better at it than I am. But here goes anyway:
I am going to assume that you don't have status with any airline. That's one way to get seat upgrades.
You COULD get an airline credit card and start banking points. (As an aside, how do you buy things like airline tickets without a credit card? How do you reserve a rental car, hotel room, etc.?) n my experience, saving points for something like Premium Economy or Business Class seats takes a loooong time. We use credit cards, but we're not heavy spenders. There are also a limited number of premium seats available for purchase by points. Just because you have enough points doesn't mean there will be availability to buy the seat you want with points.
I have recently been able to start flying Premium Economy by switching airlines. I live in Atlanta, where Delta rules. We're subject to what's known as "the hub penalty" meaning that Delta dominates to an extent that their flight prices don't have to be competitive out of Atlanta. I was a loyal Delta flyer for many years. I had Medallion status for five years up until this year. But I got to the point where I wasn't comfortable in Main Cabin anymore. I started considering other airlines, mainly United. I discovered I could fly Premium Economy to Europe with them for about a third less than I could on Delta. I also got the United credit card which comes with other perks.
United has a way to request upgrades on individula legs with money and miles. I like doing this because I can tolerate Main Cabin on the way home much more than I can on the way there on an overnight. I have never not had my PE upgrade request granted, though I have a United credit card, which moves me a little higher on the upgrade waitlist.
With no credit card, no status, you're probably going to have to just buy a Premium Economy seat. I'd start watching flights to your desired destination. Even if you don't have the date set yet. Use Google Flights and put in a destination you'd like to visit. Select routings on several airlines and then turn on the option to watch the flight price. Do this with several airlines. See if one is cheaper than the other. You'll probably notice patterns - one airline is consistently more expensive, for example. Play around with departures from different cities. That may make a difference as well.
If you need a primer on Google Flights, google "how to use Google Flights" and look for an article written within the last year or so. It really is a handy tool for keeping up with flight prices.
And don't ever be tempted to book through a third party. Third party options show up in Google Flight search results, but don't be tempted. ALWAYS book directly with the airline.