Sounds like you have some fun changes in front of you. When you put a system together each component and interconnects make a big difference in the sound you get. Changing more than one thing and/or not entirely sure of what you want can make the outcome very uncertain. If you increase the cost of most of your components significantly it is very likely to sound much better... but it is possible you could have enjoyed it even more if you had a destination in mind and carefully changed one component at a time to get there.
As far as direction. Listen to as many high end systems (that you cannot afford) to understand what kind of sound you like.
Since you are thinking of speakers, that is the logical place to start. Audition as many speakers as you can, remembering that what speakers sound like are strongly influenced by the electronics driving them. So, if you are in a store have them use the same electronics to feed the different speakers you listen to. That way at least for that audition you are listening to only differences between speakers. It takes years of experience to be able to sit down and figure out what are the characteristics you are hearing from each component in the signal chain... but with experience you can.
Then when you find some speakers that really appeal to you buy them. Stretch as far cost wise as you can... speakers get better within a brand the higher the cost, they last forever and but going cheap can limit the potential. Don’t think of changing anything else in your system for a long time... at least six months or more, save money then.
Get to know and appreciate your new speakers. It’ll take a couple hundred hours of use to just break them in. Work on positioning. Know their sound inside and out. Did this bring you closer to one of those high end systems you liked. Are all music types effected by your new speakers in the same way?
Then start thinking about replacing your integrated amp. Is your sound too analytical? Warm? Lack Details? Should you replace source or amp? Do lots of reading. Stereophile,The Absolute Sound, Robert Harleys book on High End Audio.
Do a process like above and the steps and end place will be an order of magnitude better sounding to you.
