I know the basic usage behind it, like saying "I wanna have lunch at Wendy's today", but I feel like there's a bunch of subtleties to it that I'm not really seeing yet
Like Tsuyoshi, I see no obvious connection between いい香りがする and “I wanna have lunch at Wendy's today”. I suspect you are confusing the expression 気がする with 気になる/気に入る but please clarify: it may be worthwhile sorting this out.
As for the use of する in 「いい香りがする」:
This is one of several uses of する not covered in the basic text books. In addition to its more conventional use (such as 日本語を勉強する), the verb する is also used in the following constructions to convey the following meanings:
１）To wear something, typically accessories such as マフラー, 腕時計 or ネクタイ.
eg ネックレスをする | to wear a necklace.
(する is functioning as a 他動詞/transitive verb taking を.）
2) To sense something such as におい、音、気、味
eg Nのにおいがする | to smell N
(Where する is functioning as a 自動詞/intransitive verb taking が.）
３） To cost an amount
(Where する is functioning as a 自動詞/intransitive verb which would normally take が but the amount works like adverb so が is omitted.）
Reference: Japan Times 27 August 2012