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I think that is pretty young to have a set bed time. At that age we put our LO to bed any time between 8-10 based on when she ate, when she had slept, and when she seemed tired. So maybe be flexible as to time but then do the same routine always? At that age I nursed to sleep then set her down and that worked for us for a long time. 8 weeks is really really young to expect a baby to fall asleep on their own (through training, if they just do it naturally that is another thing). I would either stick with nursing to sleep for now if it is working for you, or if you want more of a routine you could do bath, nurse, book/song, then rock to sleep.

At that age, we didn't have a strict routine (mostly because we weren't giving her a bath every night when she was that young, and also her bedtime kind of changed depending on naps), but we had a few nighttime cues. She slept in our room at night, so usually I would dim the lights and put on white noise, swaddle, nurse her to sleep on our bed (or rock, if needed), and then put her in the rock and play next to our bed. Later, we started giving her a bath every night before that. I think the cues helped her figure out pretty early on the difference between night and day. She actually never slept in the rock and play during the day (I tried, but it just didn't work), she was a baby that preferred motion, so she would nap in the swing or stroller or on someone.

That being said, she was still a horrible sleeper and after an insane 4 month regression, we had to sleep train at 5 months. So even trying early on doesn't always help. Not saying that to scare you (since your LO might do just fine!) but just to set expectations. Good luck!

I did a the same bedtime routine for every nap and bedtime (basically every time she went to sleep, except after night wakings). It was a story, sleepsack/swaddle, song. Nothing fancy.

Because newborns likely eat every 2-3 hours, you probably don't want to set a strict bedtime. For what it's worth, though, newborn bedtime is usually pretty late, like 9 or 10pm. As they start sleeping more regularly, bedtime moves up.
In the beginning, our "routine" was bath/wash, swaddle, nurse in dim light, white noise. When he got older, we put nursing/bottle before bath time and added a book after bath time.