To answer your question, buying better equipment is a huge help. Having 14 clubs you feel confident in because they're all fitted and matched is awesome and few players have that advantage. Will it improve your game to the point of making you an elite golfer? That depends on your ability. Equipment can only maximize your ability, not improve it. You'll need lessons and practice at some point or else you'll be a guy with a really nice driver who doesn't know what he's doing. The better and more consistent your game, by the way, the more you can benefit from fitting, since equipment can't stop you from making mistakes but it can help smaller mistakes be a bit more playable and good shots be better. It can also in some cases do pretty much nothing to your numbers but make things feel better, which is actually worthwhile if you have the money.
I'm pretty broke most of the time, so I don't have the option of buying top of the line stuff or making lots of changes. I make do with 1-2 year old models on sale, components, etc. I choose my equipment carefully, especially big investments like my iron set. And most importantly, I do all my own clubmaking. I just checked the specs on my clubs manually because I'd been struggling with some clubs and found I had forgotten to trim 1/2" off my 7i when I soft stepped my irons. It had played havoc with my swing since I practice with that iron a lot; I had been hitting one really heavy and long feeling club. I checked the rest of my set, and the only club I was struggling with was also the only one with incorrect specs: my 3 wood that the guy at my local store reshafted was also about 5/8" too long, despite the fact that I told the guy to go for the same length with the new shaft. I just trimmed it and it already feels much better. I play tomorrow so it should be nice.
If you want to get fitted, go to a good fitter at a good facility with expert clubmakers. Make sure they have a wide selection and are honest about how you are swinging. Obviously many clubmakers are skilled and follow directions, but they all charge money for something that's not that hard to do. I'd still rather borrow their equipment for a few minutes than actually pay them to do anything.
If you are a serious player on a budget, don't skimp on the following equipment even if you want to save money:
-Golf shoes, especially if you walk or play a lot. They make a huge difference in your swing; I forgot mine yesterday and lost strokes over it. If you are just starting out or play rarely they're not a necessity, but you need a lot of restraint and good balance to make do otherwise.
-Golf balls. They don't need to be tour balls or the newest model, but pick one model that works well like maybe a Bridgestone E5 for me as opposed to a B330 and do try out new models occasionally.
-Shafts, from driver down to wedges. They should be a good fit in weight, profile, and flex and somewhat match one another.
-Bag, unless you always ride. The comfort and utility of a good bag is worth investing in and they last years if they're made properly, but get torn to ribbons in months if they're crappy. It doesn't need to be a giant tour bag or the lightest bag
-Grips. Either go for golf pride new decade or full cord, or buy yourself an air compressor and go for pure grips. Pure seems to be the best of the traditional all rubber types and they aren't too expensive, but I still prefer cords by a long shot. They feel better to me and last much longer. Replace them yourself as often as you feel you need to, but absolutely no less than once a year. They charge actual money to do this for you at a shop, which is preposterous.
-Rain/cold weather apparel, if you play during such savage conditions. A good windbreaker is a must even if you don't play in bad weather. The umbrella is pretty overkill though.
-A little on course nutrition and hydration is good, especially playing in the elements. I always bring some, but it doesn't need to be expensive.
-If you decide to go with a pull cart, get a decent one. I don't bother with one, however.
The following are things you can always skimp on and never lose too much:
-Gloves, they are useful for covering blisters or in rain but that's about it. I have rain and winter gloves but I keep the winter gloves in my pockets the whole round and never swing with them. Wrapping a towel around the grip is better than gloves if it's really wet, though. I prefer to grow the callouses like a man and keep as much shit out of my pockets as I can. I use gloves for welding, sparring, baking, and extreme cold, not for golfing on a sunny day.
-Clubheads are overrated, especially irons. The exceptions here IMO are titanium woods and wedges, but even then it's more about fit than one brand being better. Get the right grind and loft and you'll be fine. For the rest of the set, get the proper lofts and match everything well and ignore brands for the most part. Technology is great but you can still break clubs down into GI, SGI, and players designs. Moving from one of these types to another might cause a change in performance, but within these groupings the clubs have more in common than not once you sort out the specs and shafts. The clubs from a few years ago won't spontaneously shank but they will save you 70% or so. Stock shafts are summarily lame though, even in cases like players irons where they use decent models there's probably something that will fit you a bit better. You can try before you buy if you're finicky, but I'd play almost anything with the proper specs and my preferred shaft.
-I need to be as cheap as possible in order to have a decent setup, but a big part of being on a budget is limiting what you spend on the course, not just equipment. I don't play when it costs 70 dollars and takes 6 hours, and I only take a cart if it's free or my partner insists. I also don't gamble or guzzle a case of beer every time I go out. A really cheap membership is great too, and it often lets you play a lot more golf, in addition to free range balls whenever you want. There's no way I'd have the money to play often otherwise, even if I spent no money on equipment at all.