Step voltage up or step voltage down when using batteries?

Using a step-up converter offers the advantage of being able to use partially-depleted batteries whose output voltage falls below the operating voltage of your device, but offers the disadvantages of exposing the device to harm if the battery voltage exceeds the expected level (e.g. if your 3.3-volt device operates off two AA batteries and have a 3-volt power-in jack into which someone plugs a 5-volt supply), and of increasing battery draw as the batteries get depleted (which in turn increases the risks that rechargeable batteries will be damaged, or non-rechargeable batteries will "leak" (i.e. ooze corrosive chemicals). It also, for better or for worse, will generally cause devices to continue to work normally as the batteries age, until they suddenly reach a point where they quit altogether.

Using a step-down converter offers greater immunity to input overvoltage, and will cause the voltage which is fed to the main circuitry to drop once the battery voltage has sagged too much. In many cases, this will cause the device to start working less well as the batteries age--sometimes a good thing, and sometimes a bad thing. Such behavior may be a bad thing if the device becomes useless as soon as its performance starts to degrade, but may be a good thing if the device remains somewhat useful, and if a user might not want to have to change batteries at unexpected times.

Do not put batteries in parallel. Their charge level and voltage behavior are not identical event if they are of the same model. This is due to the manufacturing process.

If you connect them in parallel, your design impose that that their voltages are equal. When the design is switched off, the battery with the higher voltage will discharge into the one with the lower voltage. If the chosen battery technology doesn't allow recharging, this energy is lost. After some time the voltage will be equal and the loss of energy will stop. But if an external disturbance appends, such as an increase of temperature, the voltage of the battery cells will differ again and some energy will be lost, again.

This is why you should never connect battery cells in parallel. And if you reverse-engineers battery powered devices, they are always connected in series.