The budget isn't really important. In the end, you want a device that reproduces sound the way you want to hear it. That unfortunately is a very personal thing. I and everyone else here can make suggestions all day long, but in the end, it's your ears and your own ideal you're trying to shoot for.
Part of the decision making process is knowing exactly what you want. I've tried a dozen of the best liked earphones suggested here, and pretty much all of them are good, some great. However, being great only goes so far. They still need to be great in your way.
If you're a person who isn't content with "decent," then you will be doing something very much like I did and am still doing. You're going to buy a few to try out, then sell what isn't the best of the group. Then you're going to buy a few more and compare, sell what doesn't come out on top for your goals. You will repeat this over and over until you find something you can really be happy with. Even then you will most likely still try some here and there to see if the new great thing can stack up with what you already have.
From a personal perspective, my two favorite earphones I've used are Yuin's OK1 bud and Westone's UM3X IEM. I still own the Yuin bud despite buying the TF10, IE8, SE530, ER4S, RE0, PFE, etc... I have yet to find another earphone that matches its realism. It is the only earphone I've used that could actually fool me into thinking the sound I was hearing was happening in real life(talking, noises, etc.) No other earphone I've used could do that, not one. The UM3X I see as one of the better offerings out there in terms of performance. I don't think it's realistic sounding, but it's one of the more detailed, dynamic, and revealing earphones available. There aren't any earphones out there that can match its dynamic range and presentation energy. It will show you things you've never heard with any other earphone. It's stage presence is king too in terms of pinpoint location and distancing, better then any other earphone I've used. However, it is very in your face putting you right in front of the singer's face. From what I've used Etymotic and Westone have been the only two companies that do stage presence to an incredible level. This isn't size but location, distancing, and sense of space.
So what about on a budget? Well, I would sort of just tell you to buy the ER4S. Grab a used pair from somewhere. It's a reference product everyone should at least try once. Despite being a 15+ year old single BA earphone, it's still the measuring stick for about everything else out there. Will you fall in love with it right away? Probably not. It's more of a product that isn't really the norm. However it grows on you quickly. Then you start comparing it to other earphones and start to pick up on the details that make it stand out among its peers: the accuracy, the balance. It just comes across more "correct." Own it for a month. Listen to a lot of music. Then start comparing it to everything else you've got. It will continue to impress you more and more. Whenever you want to let go of it, just sell it. If you bought it used, it won't drop in value and you will at least have tried one of the great reference products of the industry.
