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DSLRs usually are equipped with re-chargeable Lithium batteries. Nothing to worry here.
The two major contenders for DSLRs are Nikon and Canon. Read some detailed specs along with reviews on various models and then pick the one that suits your needs and matches your budget.
I have used Nikons for over 30 years and am extremely satisfied with their reliability and performance. Nikon lenses (film camera) that I bought 30 years ago still fit in the latest Nikon DSLRs.
Check B&H Photo as well as Newegg.
Once you buy a body of a certain make, you are committed to that brand of lenses and more bodies.