Well, if one of those tires is considered "worn" then with our horrible rain here, Stray, I wouldn't want to be driving around with that tire on. If it were summer, you might be able to hold off a month or two, but we're in one of the rainiest times of the year (spring and fall). Just look at today - yuck!!!
Hmmm, "cheap" tires? Be careful here, as you don't want to get something you'll regret. What size do you need, and what sort of budget do you have? Also, how do you describe your driving style? Off hand, the BF Goodrich T/A-V sticks in my mind as being a good choice, reasonably priced for a V-rated tire, and rave reviews to boot.
One other way you could have the battery checked is to take it to any garage about a week after you pick up the car and have them test the strength. If it's still fairly good at holding a chrage, you might be okay. If the charge is weak again, time for a new battery. Installation should either be included in the price, or done fairly cheap at any garage.
Tell us all about your new ride when you get it tomorrow! We're looking forward to hearing about it.