As a note, the takhal language is very guttural, and has several consonants not found in English. Here is a layman’s explanation of the spelling conventions/phonology (plus some sounds that exist but aren’t used in any of the names so far).

First, the sounds that sound like they could only be made by a drunk Dutchman with cerebral palsy:

“Kh.” Make an “H” sound, but press the back of your tongue up against your soft palette. If you want, put your hand on your throat. Your vocal cords should be still. If they vibrate, then try to make the sound without doing that.
“Gkh.” The same as “Kh,” but voiced, so your vocal cords should vibrate.
“Rh.” This is like “Gh,” but further back in the throat and trilled— this sound is most famous in the French language, so go for the guttural “R” sounds they do.

And, as a note the “H” sound is represented as “-gh-” in the middle or at the end of words and “H” at the beginning of words.

Now, the fancy sibilant-fricative and liquid sounds

“R.” The takhal language has no hard R like in English, instead having a trilled one. This sound is popular for its occurrence in Spanish and Italian, and I won’t ask you to try and pronounce it— I can’t either :P
“Ts.” This one is simple, and occurs in English at the end of syllables (think “cats”). Make sure those vocal cords don’t move!
“Dz.” Very similar to “Ts,” but voiced. Make sure your vocal cords DO move.
“Sh” and “Zh.” The former (“Sh”) is just like in English, but “Zh” is a voiced version of this sound, but is not distinguished from “Sh” in English spelling. In the takhal language, the voiced “Zh” is distinguished, so that’s something to look out for.

And now for the one nasal that’s different (phonotactically) from English:

“Ng” is used to represent a throaty nasal sound that is made in English when a dental-alveolar nasal (the “N”) meets a velar stop (“K” or “G”). Think “Strong,” or “Thing.” The difference in the takhal language is that this sound can appear at the beginning of syllables, so sometimes you will see something like the infamous Korean name “Nguyen.” This can be tricky at first, but it’s easy to do once you get the hang of it.

Finally, the vowels. The takhal language has a more consistent spelling of vowels (it has a lot less of them) than English. “A” makes the sound you’d expect, “Aah,” “E” makes a sort of “Ay” sound (or, in Canadian, “Eh”), “O” makes an “Oh” sound, “I” makes an “Ee” sound (“need, green, steed, feed”), and “U” makes an “Oo” sound (“roofer, poodle, chicken noodle” :]).

As for the wacky vowels, “Ë” makes a soft “Uh” sound. This sound occurs in German a lot, and is typically explained as making a very faint “R” at the end. “Ö” occurs in English in words like taught, bought, fought, and scot. It makes an “Ah” sound like “A,” but is further back in the throat and rounder. Pretend you have a potato in your mouth!

Oh, and any vowels repeated twice represent a long vowel, which means you hold the sound for twice as long! This is except for E, O, Ë, and Ö, where the long versions are represented by adding an “-h.” (Eh, Oh, Ëh, and Öh). Otherwise, it would be pretty confusing to an English/Dutch/Danish/broadly West Germanic language speaker.

Not to mention diphthongs. All that you need to know about those is that they are combinations of two vowel sounds, and they sound like they’re spelled (“Ai” is pronounced like the I in “bike,” and so on).

Now, for the pronunciation guide for all currently mentioned names:

The Yaagkhege (Yaaagkh-ay-gay)
The Nuugkhan (Noogkh-aan)
The Daanigkhal (Daaa-neegkh-all)
The Nëzhiba (Nuh-zhee-ba)
Daalag Jöm (Daaalaag Jawm)
Nëju (Nuhjoo)
Debaik (Day-bike)
Rheju (Rhayjoo)
Aad (Aaad)
Daarö of Altiil (Daaarraw) of (Aal-teeel)
Lözhan (Law-zhaan)
Ikt (Eekt)
Takhald Nëge (Taa-khaald Nuh-gay)