I think as a learner, you could just simply remember to use the word "致敬" when you wanna "say" you'd like to show your respect to somebody else. And forget the "敬礼" or "行礼" when you wanna "say" something, because these two words are mostly used to describe a behavior.

cheers! :) 我想这三个词的意思都差不多，“致敬”和“敬礼”会更相似一些。 这些词一般在比较正式的场合说，或作为书面语，生活中用得较少，有的人开玩笑时偶尔会用到。

“致敬”is usually used as a verb in Chinese, it means show respect to somebody or something For example. "我向这位英雄致敬" I showed my respect to the hero. “敬礼”(salute) is considered as a movement/action of showing respect to somebody/something. Especially used in describing something about military. e.g. “士兵们向军官敬礼” Soldiers saluted to the officer. “行礼”also means "salute", but you can use it in more occations, it means a movement/action when you meet a older/respectful person. e.g. “我向老师行礼” I saluted to my teacher. So that's it. If you have further questions, tell me as you like.