First, I would avoid credit cards if you would be tempted to spend more than you can afford, or if you're not that organized and might forget some payments. Most issuers charge very high interest rates, so if you do get a credit card it's important to pay off the entire balance each month.
If you trust yourself to do that, getting a rewards card can be beneficial. They generally aren't accepted for really large payments like mortgage payments, since merchants pay fees (including a % of the amount) to charge credit cards. But you can earn rewards on most retail purchases, airfare, etc.
I would shop around for others besides the one your bank is offering. You might prefer a card which offers simple cash back, rather than points which can only be used for certain purchases. A lot of cards offer 1% cash back, and some offer as much as 2%, although there is usually a catch like an annual fee (often waived for the first year). Others offer different %s in different merchant categories.
Bonuses like the 20,000 points your bank is offering are also common, so shop around. Some cards offer cash bonuses, generally in the range of $100-400. But you have to consider whether you will spend enough in the first few months to qualify for those offers.
If you're very organized, you might eventually consider getting several cards - one with high rewards at gas stations, one for grocery stores, etc. You can also get a nontrivial amount of money from the sign-up bonuses. But keep in mind that it would be more trouble to pay off several cards each month (although it can be automated), periodically check each one for fraud, and so forth.