The definition of the words "battery backup" is literally a battery designed to supply backup power for a device. Therefore a battery backup by definition does not include a pump, float, or anything other than a battery which is typically charged off an outlet.
You cannot "just go out and buy a battery backup and plug your pump into it" unless your pump happens to be a DC pump designed to run on the voltage supplied by your battery backup. Most house sump pumps are AC, designed to run on 120 volts AC, and they will not run on a battery. If you purchase a battery backup with an inverter designed to convert the DC power produced by a battery to 120 volt, 60 cycle AC power, you can run your sump pump off that. The run time will depend on the size of the battery.
I don't understand your question about "Any recomendations for a back up costs, etc. ?". If you mean should you spend the money on battery backup, that depends entirely on the risk you are running if the sump pump fails during a storm and your basement floods. Only you know what you are planning to put in the basement. Consider the option of a backup generator, which would cost considerably more, but could power your house in the event of a power outage.
The whole question of backup is pretty complex, because there are many potential points of failure in the system. For example, if the pump fails, your battery backup means nothing. So should you have two pumps? If the float fails, your pump may not turn on, so should you have two floats? The battery might fail due to lack of charge, or deterioration of the cells - should you have two battery backups?
You really need to think through the entire system, and decide how much risk you are willing to tolerate. I have a sump pump in my basement, which has worked for 18 years. It is a Barnes industrial pump, cost about $400, but well worth it. My previous low grade sump pump quit in the middle of a storm, I got 18 inches of water in my basement. I do not have a battery backup, nor do I have a generator, my defense is that there is nothing valuable left on the floor in the basement.