I apologize if this has been answered many times before, but I couldn't figure out how to search the forum for it. I'd like to learn the idiom a Thai would use to express the thought, "On the one hand ..., but on the other hand ..."

How about:

แต่ว่า (tae (l) wah (f))

from http://th.w3dictionary.org

def:[เป็นคำเชื่อมทำให้ความกลับกันหรือขัดแย้งกัน]
sample:[เขาได้แต่งงานกับคนที่เขารัก แต่ว่าเขาไม่มีความสุขเพราะผู้หญิงคนนั้นไม่ได้รักเขา]

The sample given here is : He married the woman he loved, but he isn't happy because she doesn't love him.

However, I think you can also use it quite handily in the way you suggested in the OP

I don't know of any idioms, but to say what you want to, the words ถ้า (Taah - If) and แต่ (dTair ~ But) or แต่ว่า (ว่า = Waah ~ That) would come in handy.

โดยเฉพาะ (Doi Cha Powh ~ Especially) and เพราะว่า (Prowh Waah : Because that) are also words which I often use when having this type of conversation lol

ดีกว่า (Dee Gwaa ~ More good (i.e. Better), you can replace the ดี (Dee ~ Good) with any Thai word e.g. แพงกว่า (Pang Gwaa ~ More expensive) is also another word which would be good for this type of conversation. You can also also use the final particle ซิ when speaking to make the sentence kinda, more persuasive or suggestive that you think something is a good idea (Although I'm still not confident using this particle yet lol, as need to hear it used more often first)

But of course, if you know all these and are just looking for an idiom, sorry I can't be of much assistance there (And my wife said there probably is an idiom, just she can't think of any off the top of her head).

On the one hand ..., but on the other hand ...

"On the one hand" although common, is redundant.

on the other hand = conversely = ในทางกลับกัน nai taang glap gan

One the one hand he is rich but on the other hand he is ugly. = He is rich but conversely he is ugly. = เขารวย แต่ในทางกลับกัน เขาน่าเกลียด

on the one hand: ในด้านหนึ่ง S + V

on the other hand: S + กลับ(ต้อง) + V, ในทางกลับกัน S + V

I have used reu mai ko หรือไม่ก็ for "on the other hand".