だけど details

Examples

but; still — pushes back on, or qualifies, what was just said

行いきたかったんだ。だけど、バイトが入はいっちゃってさ。

ikitakatta n da. dakedo, baito ga haitchatte sa.

I did want to go. But then I got put on a shift. casual

「この店みせ、安やすいよ。」「だけど、駅えきからちょっと遠とおくない？」

"kono mise, yasui yo." "dakedo, eki kara chotto tookunai?"

"This place is cheap, you know." "Sure, but isn't it a bit far from the station?" casual

お誘さそいいただいてとてもうれしいです。ですけど、その日ひは先約せんやくがあるんです。

o sasoi itadaite totemo ureshii desu. desukedo, sono hi wa sen'yaku ga aru n desu.

I'm really glad you asked me. It's just that I already have plans that day. neutral

でも
でも rides along into polite speech unchanged (でも、そうですね), so it is the safe objection at any level; だけど carries the plain だ and stalls in polite talk, where ですけど takes over — which is why でも, not だけど, is what you say back to a teacher or a senior colleague.
だが
The けど／が split survives the move to the head of the sentence, so だが is plain-style WRITING (editorials, narration, commentary) while だけど lives in speech and messaging and turns up in print mainly where the writer is imitating a voice.

けど
One clause boundary apart — けど welds onto the end of the clause it concedes (安やすいけど、遠とおい), while だけど is that same けど with the copula frozen onto its front, opening a new sentence about one already closed; a learner who hears だけど mid-turn and looks for the first half inside the same sentence will not find it.