@Ren
This is going to vary based on how hard you intend to hike, the conditions typical of your region, your experience level, etc., so I’m going to give some general advice assuming you’re hiking in roughly freezing temperatures.
Firstly, be safe. Don’t overestimate your abilities, and don’t underestimate mother nature.
So, if your budget is limited, winter will present issues. There are a lot of ways to cut corners in warmer months, but winter can go badly if you’re unprepared. I would advise that you start small with trails that have an easy way to get out and back home should things go south.
You’ll need layers, and you’ll need to experiment to find the right layers for your body’s temperature regulation. You need baselayers, into insulation layers, into weather layers. Everyone needs a different layering scheme. I overheat and sweat easily, so I use very few layers while active. You want to avoid sweating. When you go static, you want layers available to retain the warmth you’ve generated through activity. When you head off again, strip the layers back down.
Baselayers can be synthetic (cheap and durable) or merino wool (expensive but insulates when wet). Insulation can be fleeces (cheap and durable, quick-drying) or down (synthetic - cheaper and more resilient, or genuine - expensive and more fragile, provides more warmth). Weather layers can be softshells (weather-resistant) or hardshell (weatherproof).
You need merino wool socks. Smartwool and Darn Tough are both great. They fit differently, so try both on.
Boots? I’d recommend leather, then, but it’s not cheap. It may be well out of your budget. It is, however, best for cold weather because it naturally insulates to a degree without artificial insulation, making it a more versatile option. Leather boots are often resolable, and the upper of a good quality leather boot will last many years.
Leather boots: Asolo TPS 520 GV Evo, Scarpa Kinesis Pro GTX, Meindl Comfort Fit Hiker, Extreme, or Hunter, Hanwag Tatra, Hanwag Lhasa II, Hanwag Tatra II, Kenetrek Mountain Extreme, Zamberlan Vioz Lux GTX RR, Zamberlan Vioz GTX.
Those are some of the best ones, but they’re all pretty pennies. The alternative is military boots, preferably leather with a Gore-Tex membrane, but a high-quality one without Gore-Tex can still be waxed to be highly water-resistant.
If you can’t afford or find that, you can look into synthetics with insulation if you’ll be static at freezing temperatures, but this again depends upon your body’s temperature regulation. You may be fine with a synthetic without insulation at freezing with the right socks, like me, or you may need insulation. It just depends.
You need a pack to carry your layers and the 10 essentials.
https://www.rei.com/learn/expert-advice/ten-essentials.html
Scale these items up or down based on budget and how hard you intend to hike, keeping in mind that it’s your hike. You don’t have to scale a mountain in the winter to prove anything, and you certainly shouldn’t have that as your starting goal. Just get out there and have fun, but be safe.