I am guessing you've already managed to export the texture as a BMP or PNG file and it's 8-bit (indexed). If you haven't and don't know what the latter means, I'll explain. But anyway, let's say you're using GIMP and you got your texture with its palette.
In GIMP, you'll go to Windows -> Dockable Dialogs -> Colormap.
You will see something like this:
On the left side, you can clearly see the palette and its 256 colours. From then onward, changing the palette is quite simple. You can right-click on one of these colours and click "Edit Color".
There's just one problem: if you want to change multiple colours, like, all the blue ones into red, it's gonna be tedious because you'll have to modify every colour that looks like the one you wanna replace (there are multiple blues in my case, and multiple greens, kinda hard to put my finger on the line there). For that, you can use fresco's suggestion. Once you import the texture into your image editor, you switch the mode to RGB, which will get rid of the 256-colour palette and let you do whatever you want.
If you want to change colours that way, you can select an area, and go to Color -> Hue-Saturation, and play with the hue setting:
There is no easy way TBH. It all depends on what exactly you wanna do, and some types of edits will be simpler than others. You gotta practice to get the results you need, maybe for a couple of months or something. There will be pain and frustration along the way, but it's worth it IMO.