I'm not sure which cards can transfer points to Amtrak.
Getting a credit card doesn't mean you have to owe money on it. I've had credit cards for almost 40 years and I don't think I've ever paid a penny of interest; I've always paid the bill off when due, or sooner.
The real juice in the credit card game is signup bonuses. Let's say you each spend $500 a month. The signup bonus on a Chase Sapphire Preferred will be equivalent to what you'd get on seven years of spending.
I don't know your credit scores, but getting a new credit card generally dings your score very little, and they do come back. I often say they don't engrave your credit score on your tombstone - in your case I'll say instead they don't put it on your transcript. You can check out a site like CreditKarma for an idea of your score, or go to myfico.com and sign up for the free trial (don't forget to cancel before the trial period is over).
The only problem is that some cards require a minimum spend to get the bonus. For example, CSP requires $3000 in three months. There are some tricks and techniques for meeting that - just as an example, you can charge 6 (or more) months in advance of your wireless and cable bills. Basically, you don't want to get cards faster than you can meet the spend requirement without buying stuff you wouldn't buy otherwise.