I have taken my bicycle on both Amtrak and Greyhound.
1) It is always (or almost always) late. I'm not talking about running 15 minutes late ... I'm talking about 8 hours. Just something to plan for.
2) Look at the Amtrak website for the station where you plan to start your trip and the station at your destination. Some Amtrak stations allow you to check luggage ... some don't. Some allow bicycles ... some don't. More don't than do.
3) Sacramento was one station that allowed bicycles ... and I have to say that they were very good. I got there about noon, and my train was scheduled to leave at 11 pm. I didn't want to haul my boxed bicycle, laundry bag full of panniers, etc., and my Carradice carry-on all over downtown Sacramento while I waited, and I didn't want to sit at the station guarding my stuff. So I went over to the baggage counter and asked the attendant if I could check my bicycle that early. No problem at all. I checked both my bicycle and laundry bag without any difficulties. Then I asked if I could also leave my Carradice there for a few hours and pick it up a bit later. I thought that would really be a long shot ... but for $5 it was no problem at all. I was free to roam around downtown Sacramento before finally returning to the station and retrieving my Carradice in the evening. Unfortunately the train didn't actually arrive till about 7:30 am, so I had to try to sleep on the hard benches of the station all night, but at least they were really good about the luggage!
4) Eugene, OR is another station that allows bicycles ... and here's something you might want to make a note of. When the train finally pulled into the station (now about 12 hours after the time it was supposed to arrive), everyone piled out and rushed to the arms of their waiting loved ones ...... everyone but me. I rushed out and bolted for the one and only wheeled cart in the place. I spotted it on the platform as we pulled in. I had it loaded with my bicycle and laundry bag, and everything by the time the rest of the passengers detached themselves from their families and went looking for one. I got a lot of very dark looks when they realized I had the one and only cart available ... but who cares!! I was on the ball ... they weren't! Plus I had about twice the luggage as anyone else with my boxed bicycle. So, when you pull into the station ... keep an eye out for the rare and elusive luggage carts!
1) I wouldn't recommend riding any distance on a US Greyhound. If you were travelling in Canada or Australia, I wouldn't have a problem recommending Greyhound ..... but they are scary things in the US <<shudder>>
2) You might be able to check your stuff early ... I was able to check my bicycle and laundry bag in Eugene ... but I don't think that you can do that everywhere. Greyhound also provided lockers where I could store carry-on stuff so I could wander around downtown Eugene freely while waiting for my bus.
3) The luggage carts are also few and far between at Greyhound stations. If you can find one, grab it immediately!!
4) Sometimes Greyhound can be a bit odd about bicycles.
--- At some stations they seem to have no problem at all with the idea of loading the bicycle up with the rest of the luggage.
--- At some stations the baggage handlers seem offended by the idea of having to load a bicycle box and they can be a bit rough with it. I will usually wait outside the bus, watching what is going on, until my bicycle has been safely stowed underneath. On one bus change (in the US they seem to need to off-load all the passenger and luggage and put them all onto a different bus every 100 miles or so), the guy had tossed my bicycle off the bus (it was airborne!) before I could get close enough to stop him ... but I can tell you, he didn't toss it back onto the bus!!
--- And at some stations someone might come along and tell you that you have to take your bicycle to the cargo area and send it that way ... meaning that it will arrive 2 or 3 days after you do (but perhaps in your case, you might want your bicycle to arrive a few days later). Sometimes they might even threaten to charge you extra, over and above the extra you've already paid for the bicycle. I discovered that the way around this is: A) when you are standing around outside the bus waiting for everything to be loaded, make friends with the driver. If you're really nice about it, often the driver will have no problem loading a bicycle on. B) stand your ground (politely, of course) ... especially if they want to charge you extra. Also keep every shred of paper they give you along the way as proof that you've paid all the dues for the bicycle and that the bicycle is allowed on the bus with you.
5) Sit near the front. Not only will that keep you out of the way of the psychos who are brawling in the back of the bus, but it will give you a clear path to get out the door to keep your eye on what's happening with your stuff.
Just a few tips.