Unlike airlines, which often publish certain fares in certain buckets only valid on certain days of the week (e.g. low fares on Tuesdays, Wednesdays, and Saturdays, which are days that are typically not business travel days), Amtrak offers the same fares in the same four buckets regardless of date of travel. The only thing that affects the price is how many seats have already been booked, which determines which buckets are full and what fare bucket they are currently selling at.

If you book early enough, or if you are lucky enough to pick travel dates which are relatively lightly booked, you can get the lowest fare, regardless of what day of the week it is.

Since Amtrak's long distance trains (like the Coast Starlight) generally target a leisure audience, the day of the week isn't going to make as much difference (although weekends may trend slightly busier). The Cascades, though, are used by business travelers and so avoiding Mondays, Thursdays and Fridays may help. (Then again, I haven't noticed much variation in Cascade fares, so they may have a simplified fare structure of only a couple of buckets, and the difference between them may only be $10 or so per segment.)

One final note: avoiding credit card debt is noble and commendable. However, establishing a good credit profile as early as possible will help you greatly down the road, as when you go to apply for a car loan or mortgage, they look at the length of time your accounts have been active, among other factors. So if you get a credit card and can discipline yourself to use it responsibly (pay your balance off COMPLETELY at the end of the month or--my preference--every few days [making it virtually a debit card]), you'll end up better off. Even if you just get the card and don't use it (put it on the top shelf of a cupboard or, to really discourage yourself from using it, freeze it in a jar of water in your freezer), having the account open will help you in the long run. Of course, you may not be able to get the AGR card immediately if you have no existing credit history, but you should start off with a student card or a secured credit card as soon as you are able in order to begin your credit history.