There are a lot of artists online who use Copic markers or a mix of Copic, other brands and even other media like pencils to create texture or white ink for highlights. Often a lot of cool images you see online are actually a mix media of markers and other materials.
Lets start at the beginning though... You mentioned markers you used to use had uneven coloring. Most basic markers or kids markers will do this. Markers like Copic, ProMarkers etc are alcohol based, they don't dry as quickly on the page, allowing you to create a nice block of color or to blend two colors together.
You need to be careful when coloring though, your color edge can dry and then you'll be left with a visible seam in your color block. Always follow the wet edge of color adding to it each time before it dries, and even go back over your colors so you get a solid block of ink down if that's what you're looking for, as the first pass might be a bit uneven depending on your paper type.
This video will help explain what I mean if you're like me and like to learn visually: https://www.youtube.com/watch?v=PZpBpAdQliU. That YouTube channel actually has a good amount of information on Copics that may help you get started.
So, a great way to start is to practice blending colours together if that's your style. I suggest diving in and using the markers to get a feel for them, I used promarkers for a long time and it took be a little while to get used to them at the beginning because they were very different from what I was used to (Faber Castell / Crayola etc).
Paper is always an important factor to me. Specifically designed marker paper (Daler&Rowney, Letraset, Copic etc all make it) helps maintain control over the ink flow. For me anyway, drawing on regular paper or cardstock meant going too close to the edge of my ink outline risked it leaking beyond - Marker paper was amazing for this and reduced bleed a huge amount for me. It may all depend on your style and what kind of pieces you want to make. Don't just stick with one paper type, try a few and experiment, you'll probably find you prefer different papers for different styles and quickly choose a favorite.
Aside from blending another great way to use these types of markers is by layering in blocks. Rather than constantly trying to blend seams together, you can make great pieces by being hard and direct with your strokes, or make pieces that feel dynamic and full of motion by being loose and rough. Terrance_Unchained on Instagram is a great example of this. https://www.instagram.com/terrance_unchained/?hl=en. He uses chisel tipped copics a lot of the time to layer down blocks of color. If you follow the link in his profile he does a tutorial on his style. He also uses other inks and pencils etc to highlight and texture areas like I mentioned at the start. One of my favorite techniques of his that gave me an "omg that's genius moment" is when he's finishing up a picture, he'll go over the whole thing with the one color to tie all the tones together - I've yet to try this myself but it looks really cool.
Hope this all this helps! Good luck with your Marker adventure!