What would be the procedure to terminate the services with my management company in Tampa Florida?

I asked them for a copy of our agreement and they told me. That I only have a hand shake agreement with them.

Because I found a company that would only charge me 50% of the fees that my present company is charging me.

Thank you

Manage what? A home when you aren't there?
You terminate the way you started - except instead of a handshake, a wave. Just deny that it was anything but month to month. If they sue, they will lose.
From now on, in writing, OK? Half price sounds just as ominous and your handshake deal. You need a specific list and terms.
And if they are going to have keys to your house... bonded employees. But bonding costs more.
So you could be asking for trouble, we don't know!

You don't remember if you signed a contract or not? Hummm

If they say no, I would give them written notice to stop services perhaps a 30 day notice, and then notify all tenants or people involved of the date of the change.
Nothing is free, so for 1/2 the cost, either you were being cheated, or the new company is not going to provide some services.

Manage what? A home when you aren't there?
You terminate the way you started - except instead of a handshake, a wave. Just deny that it was anything but month to month. If they sue, they will lose.
From now on, in writing, OK? Half price sounds just as ominous and your handshake deal. You need a specific list and terms.
And if they are going to have keys to your house... bonded employees. But bonding costs more.
So you could be asking for trouble, we don't know!

They are managing a revenue property (condo)
They do the collection & deposits.
They take all calls from the tenant.
They schedule repairs if there is need

If you only have a handshake agreement then it can be terminated the same way. Contact the new company you are considering. Find out from them how much lead time they require to take over. Ask them to send you a contract with the terms spelled out. Have an attorney review it. Once you are ready to sign, you do as Chuck suggested; send them a notice that their services will be terminated as of x date and to send you a final invoice. Then inform the tenants that, as of y date, they should make payments to the new firm and use the new contact information.

... I would give them written notice to stop services perhaps a 30 day notice, and then notify all tenants or people involved of the date of the change. ...

First, get the new company to provide you a signed contract. The contract should be to take effect upon termination of the old unwritten contract (date as specified below).

Then OP should send a letter to the management company, as well as to the tenants. All letters should be certified mail. The letter should say that, as of the date of receipt of the letter, all rental payments are to be made to the new company. The important thing is that the tenant(s) will be put on notice that they are to now make payments to the new company, and the old company will know that it will not be entitled to any more fees.

The letter should also require the old management company to account to OP for all moneys received, charges incurred, and moneys paid. Thus, for example, if rent is received before the letter is received, OP will know about it.