Hi Alex,
Lots of good advice so far. If this is your first property and you're currently renting, your best bet might be to use an FHA loan (or VA loan if eligible) to buy a 4 unit property and live in one of the units. It will give you hands on management experience and a place to live (hopefully inexpensively with tenants covering much of the mortgage). Even under a 580 score, you can use FHA to purchase a 4 unit with 10% down. Would any family members consider gifting your down payment, or can you set some money aside from your salary to put together a down payment?
In regards to the collections and late payments, the sooner you can get those paid off or settled, the sooner your score can start recovering. Simply paying off the debt will not remove it from your credit report, but the older the derogatory information is, the less impact it has on your score. After 7 years, it should fall off your report.
When talking to creditors, it would be best to NOT mention that you want to clean up your credit so you can purchase a home. If they know you need to deal with the account quickly, you'll have less leverage.
The very best thing you can do right now is talk to a good loan officer and have them pull your credit. It can even be a soft pull with all three bureaus. They can guide you in the next steps once they have the full picture (how large the balances are, how recent the late payments or collections are, types of accounts such as medical debt, installment debt, or credit cards only, which accounts have been disputed, etc.)
Debt settlement companies usually charge a monthly fee, so the longer they take to resolve your accounts, the more they can collect in fees. Just something to be mindful of when monitoring how they are handling your accounts.