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I would join DCU and apply there. Also most communities have good local credit unions. You should share your full credit information as well as his if you want an indepth answer. All credit unions have online application processes so that part is very easy.
You should establish credit asap, apply for a credit card, even if you have to do a secured card. Capital One is a great first card to go after. Establish a couple of cards and use them and pay in full each month. Everyone needs a credit score. Establishing yourself for just a few months will put you in a good place for a car loan as well as a mortgage. Failng to establish a credit history will make you a liability not an asset for a car or a mortgage. Many times one partner or the other becomes reliant on the others credit, this isnt a great idea. We work hard to build both of our profiles so when we are going to get qualified our case is stronger than if it was just me or my wife who had established credit. Currently hers is better than mine because I have some student loan lates so her credit history has helped us quite a bit these last couple of years.
Used versus new is always a debate. You can finance either pretty easily if the loan to value ratio and other credit indicators are acceptable. Cars that are the current or previous model year are generally treated as new even if they are previously owned. The key here is buying smart, dealers love to pile on all kinds of add ons that you mostly don't need like an extended warranty on a car that already has a great warranty and face it cars these days are extremely reliable. All new cars have at least 3 yrs 36,000 miles bumper to bumper and 6 year 60,000 miles powertrain covered so paying 3-4k upfront for a warranty doesn't make much sense for most. If you can avoid having that stuff piled on the deal and buy at a fair price (between invoice and MSRP, closer to invoice or below is best) then you will be fine for your loan to value.
I agree with the post above, avoiding a co-signer is ideal but if you need to establish credit and you can get a much better APR it is probably worth it for this first purchase.
I too have plenty of cash and have lived for many years purely on cash but most of us don't have enough to pay for a house with cash and frankly even if you do it might not be the best place to put that cash. Building credit provides you with options and having a totally fresh start puts you in a perfect position to have perfect credit which means finacing cars at zero percent so you can put that cash to work in an investment that pays you instead of in a depreciating asset.
You'll do great with credit because you have learned to live without it. Those who get in trouble are generally those who become too reliant on it for day to day expenses.