Tense-neutral vs past-only
然后 is the generic sequencer that works in any tense. Recipes: 先切菜,然后放油 (first chop the vegetables, then add oil). Future plans: 明天开会,然后吃饭 (tomorrow we meet, then eat). Past: 昨天开会,然后吃饭 (yesterday we met, then ate). 后来 is past-locked: 我去了北京,后来去了上海 (I went to Beijing, later went to Shanghai). You cannot say 明天后来去 for future; it is ungrammatical.
早上我去跑步,然后吃早饭,然后去上班。
Zǎoshang wǒ qù pǎobù, ránhòu chī zǎofàn, ránhòu qù shàngbān.
In the morning I go running, then eat breakfast, then go to work.
Step-by-step vs time-elapsed
然后 implies step-by-step sequence with small or no time gap between steps. It is the "next step" marker. 后来 implies time has passed between the first event and the second, often significant time: days, weeks, years. 先开会,然后吃饭 (first meet, then eat, tight sequence). 我大学毕业后去了北京,后来又搬到了上海 (after college I went to Beijing, later moved to Shanghai, big time gap).
Cousin: 接着 (jiēzhe)
接着 (jiēzhe, next / right after) overlaps with 然后 but adds the sense of "immediately following". 接着 often emphasizes continuity: 他说完话,接着就走了 (he finished speaking, and right after left). 然后 is neutral-sequence; 接着 is tight-sequence. For recipes and instructions, both work; for narrative continuity, 接着 carries slightly more flow.
先讨论方案,接着决定下一步行动。
Xiān tǎolùn fāng'àn, jiēzhe juédìng xià yí bù xíngdòng.
First discuss the plan, and right after decide on the next action.
Position: both go before the second clause
然后 and 后来 both appear at the start of the second clause or sentence. 我吃饭,然后休息 (I eat, then rest). 我去了北京,后来又去了上海 (I went to Beijing, later I went to Shanghai). You can also put them as standalone connectors between sentences: 我去了北京。后来我又去了上海 (I went to Beijing. Later I also went to Shanghai).
Combining with 以后 (yǐhòu)
以后 (yǐhòu, after) is a general "after" marker that works for both past and future. It often pairs with specific events: 下课以后我去吃饭 (after class I will eat). 以后 can replace 然后 or 后来 in many sentences but adds a different feel. For pure sequencing "then", 然后 is best. For "later on" in past narrative, 后来. For "after a specific event", 以后.