It sounds like flexible points would be more useful to you than miles that are tied to a specific airline. There are several flavors of flexible points (including Chase Ultimate Rewards, American Express Membership Rewards, and Citi ThankYou Points), but I think for the budget-minded traveler who wants to keep things simple, Chase is the way to go. Chase has an excellent travel portal where you can use your points to book flights, hotels, etc. You'll want to get either the Sapphire Preferred or the Sapphire Reserve. Both currently come with a 50,000 point bonus.
The Reserve has a much higher annual fee ($450), but it's offset by sweet benefits including a $300 annual travel credit, extra points on travel and dining expenses, and a Priority Pass card that gets you into airport lounges. Also, with the Reserve, your points are worth 1.5 cents per point in the travel portal, whereas they are worth 1.25 cents per point with the Preferred card. So, for example, if you find a flight on the Chase portal that would cost $500 cash, it would cost you about 33,000 points with the Reserve and 40,000 points with the Preferred. A lot of people find that the Reserve is worth keeping despite the high annual fee, but it comes down to whether you can make good use of the benefits and the higher-value points.
As for meeting the minimum spending requirement, there are a number of ways to get this done "artificially," including by opening a bank account that allows you to fund it initially with a credit card. You can also load your Amazon balance with a credit card (though you would then be floating the money for a period of time). But I think buying $500 Visa gift cards is the easiest way. You can liquidate these via money order, but this will require a bit of research on your part and is easier in certain areas than others. You could also ask friends or family to help you by buying these cards from you, and they can use them for their regular spending.
If you decide to apply for the Reserve, feel free to PM me if you'd like to use my referral link! I realize this post sounds like a Chase ad, but I seriously love my Reserve card. I do manufactured spending on a Chase business card that keeps Ultimate Rewards points flowing into my account.