There are no preferred forms for the notice to vacate. I suggest posting it on the door and mailing by certified and regular mail. The time starts from receipt, so send as many ways as possible. Something like, "I purchased your property at the foreclosure auction on ______, 2022. This is your notice to vacate the premises within ten calendar days. Failure to vacate in a timely manner will result in loss of any redemption rights and subject you to a lawsuit and a request for damages for unlawful possession." Put your contact information at the bottom so if they want to redeem they can contact you. mostly, you don't want them LATER saying they would have contacted you to redeem but didn't know how to get hold of you.
You can buy them out rather than using courts. The going rate is $2,000 cash for keys if they vacate in three weeks, but they also have to release any redemption rights to you.
If there are no problems (e.g., a claim the foreclosure process was invalid for some reason) then an ejectment lawsuit will cost around $500 for the lawyer and $200 for the court filing fee and $50 for a default judgment. That will take 30 calendar days from when you obtain service. I recommend a private process server for the speed and the ability to instruct the process server to video themselves serving the papers. That way, the defendant cannot later claim they were not served and possibly get a default judgment set aside. Private process server will cost around $75. if they file a generic answer and force you to prove your right to an ejectment order, your lawyer can file a motion for summary judgment. That usually costs a couple of hundred dollars extra. You can nail this stuff down when interviewing lawyers. The motion for summary judgment tells the court "There are no fact issues in dispute, maybe only questions of law. Because of that, there is no need for a trial so a trier of fact (judge or jury) can decide the correct facts. So, here are the facts, they are undisputed, now please rule as a matter of law that we win." That usually takes about three to four weeks. Then the ejectment order has to be final and non-appealable, which takes 42 days. Assuming there is no appeal and no post-judgment motions, you can then get your turnout order. That takes a couple of days. In most counties, you can them make an appointment with the sheriff's department for a deputy to accompany you and keep the piece while your movers put everything out on the street and the owners are escorted out. If they resist, then you go back to court and ask for them to be found in contempt of court. that takes about a week. If you are in Jefferson County, then the turnout orders have a backlog of several months because of there being too many ejectments and evictions to administer, and not enough sheriff's deputies. All other counties it is pretty fast.
For the above reasons, it usually makes sense to negotiate cash for keys.
By the way, it is not usually the borrower you have to worry about redeeming. It is the borrower selling its redemption rights to another investor, who the redeems and snatches your deal away from you.
If there is a redemption, you are entitled to repayment of the auction bid price (not the price you paid the bank, if you bought it from them after the auction) plus casualty insurance premiums, plus the value of any permanent improvements or repairs made after the auction, all at 7.5% interest.
If you have to eject, you are also entitled to damages equal to the fair rental value of the property from date of demand until delivery of possession, but borrowers are usually judgment-proof, so don't pin a lot of hopes on that being collectible.