Good morning.
As you may have surmised, this is a pretty complicated area. The wiki
http://www.dwarffortresswiki.net/index.php/Tilesets is a good source of help here, as with most things DF-related.
If you started doing this stuff manually (which is highly recommended), it can be fairly straightforward (if somewhat tedious) to figure out what's causing the problem. On the other hand, if you let automatic installers do the work, things can get messy quickly.
In general, there are four areas related to display in DF.
1. The data/art folder holds your character sets (what's used for text and most other display elements).
2. The init.txt file (in the init folder) controls WHICH character set you are using for each display mode (there are 4). The init.txt file also controls whether you are using graphics, or only characters.
3. /raw/graphics contains the definition files for any creature/sapient graphics you are using. The actual graphic files referenced in 3 above are generally stored in sub-folders under /raw/graphics (like /raw/graphics/creature).
4. /raw/objects controls how many (but not all) in-game items (such as creatures, plants, etc.) are displayed.
As you can see, there are quite few settings and files involved, and it can be very easy to cross the wires and have the wrong data referenced in one or more of the files.
Since not all character sets have the tiles in the same order, for example, changing from one character set to another may cause total havoc in your display.
Virtually all mods can be made to "play nice" with each other, but the process can require a good deal of tweaking. I personally play with MANY mods together, and after several rounds of tweaking I think I've got all the major bugs squished...the ones I've found at the least.
To get things working right, it is essential to determine exactly what files are added/changed by each of your mods. With that knowledge in hand, you can determine where they are conflicting, and make the necessary adjustments. In most cases this is pretty easy, but if two mods BOTH overhaul, for example, the tree character settings, you'll have to manually adjust the data before it will work properly.
Specific recommendations are highly personal. I personally prefer the classic Dystopian Rhetoric graphics and a fairly low-impact character set (Flying Mage or Shadow Lord). Others prefer Mayday and/or Sphr and its descendants (like Ranting Rodent, arguably the most complete graphics set going).
The easiest way to figure out which is best for you is to create a number of different DF copies that are all identical EXCEPT for the character set/graphics you have installed on each. Then, gen up a world, start a game, and get it to a point where you have a decently-sized fortress finished. Take that save game and open it up in each different copy to see how they all look with the same environment. Choose the one you like, and delete the rest.
Of course, by that point you may be wanting to make your own character/graphic sets, since you can't find JUST the one you want. Welcome to DF.
Rochndil, working on a character set right now...