If you are someone like me where you play more as a hobby, but have spoiled your ears with some of the amazing gear an instruments out there in the world. You know the struggle of not being able to really improve your tone. There are so many variables that affect your core tone, there of course is the actual instrument you are using along with your amplification. When taking a further, more in depth look there are other factors such as your intonation, what kind of strings you are using, what gauge strings, the pick ups in your instrument, what amp, what cabinet, tubes or solid state, , effects, length of signal chain, even what kind of cables you use. The list can drag on and the deeper you dive into the specifics of tone the more sensitive your ears will get to these different factors.
One of the first things you should consider is budget, of course a bigger budget will let you buy better quality gear and more of it, but if you have a small budget you will want to focus on more of the basic essentials and slowly work to adding on to your rig and improving your tone.
Replacing pick-ups is a great way to improve your instruments tone. If you have a more affordable instrument, chances are that one of the things that made it affordable is the pick-ups. Replace pick-ups can make your guitar or bass sound like an entirely new instrument for the fraction of the price of an entirely new instrument.
Effects pedals are another way to supplement your tone, they can also dramatically change it. There are so many options and types of pedals at so many different price points; of course it all comes down to the sound that you are after. To solve the problem of budget, while trying to get the most out of what you can afford, TC is a great option, their Tone Print technology can provide a player with so many different effect options with in their respective pedals. There are also great budget pedals such as Tone City and Moore, which specialize in micro-pedals that still can keep up with full sized pedals.
My last tip and probably greatest tip for the issue of "tone on a budget" is found on your laptop. The amp is something that essential for performing, it also can be the most expensive part of a player's rig; it is also the loudest. Like many others I live at home with my family and when at school have neighbors to worry about, so amplification is not something that I can really enjoy. When playing I plug into an audio interface that runs into software on my computer. The interface converts the signal from my guitar/bass to a digital signal that my computer can read an run into a wide rang of emulated effects, amps, and cabinets; all depending on the software that I use. Garageband has served me well for a very long time and I really recommend starting there for anyone who owns an Apple computer, since it comes free with iOS. I recently purchased BIAS FX which has been one of the best decisions I have made, their standard version only starts at around $60.00 and provides you with a wide range of emulated amplifiers, cabinets, microphones, and effects. There are many expansions available and the best feature is their tone cloud. This allows you to access an enormous library of user made pre-sets that you can download and edit.