Modern vape machines come in two main battery styles, internal battery devices and external battery devices. Internal battery devices have a built-in battery that cannot be removed safely. External battery devices have replaceable rechargeable batteries. Both of these styles have different safety requirements for proper operation.
Internal batteries are the most common style for a starter kit type device, so most people will probably use one of these. An internal battery will typically be recharged through a MicroUSB or USB C slot on the device. These devices are often rated to accept both normal charge and fast charge, without causing damage to the battery. Think of your internal battery vape as operating like a mobile phone battery. Mobile phone batteries are safe enough to carry around daily and so are internal battery vapes. The biggest danger comes from drops that may damage the internal battery causing the battery to vent. This can be noticed by the device heating up or ejecting gasses. Leave them alone if this happens and bring it into a local vape shop, when it finishes venting, for inspection and advice.
External battery mods can be opened up to remove and replace the batteries. Because this method is a lot more complicated than simply plugging a device in there are a few more issues that can pop up. The batteries used in these devices are typically wrapped in soft plastic with an insulator ring around the positive end. This wrap and the insulator ring are your main safety features and should be looked after. A single battery device or a device with balanced charging can be charged internally but an external charger is the most common way of charging. Please make sure to purchase one from a vape store or take yours in to be inspected, standard AA chargers will not work and may damage your batteries. When using an external charger please make sure your battery is of the correct size, if it is too big the plastic wrap will tear easily and you may even get dents on the ends. Always inspect your battery after removing it from a vape or charger as tears in the wrap or insulator ring can lead to arcing batteries or a short circuit. Another thing to note when inspecting batteries is a build-up of black gunk under the battery wrap. This could indicate that it has taken some damage and could be leaking internal fluids. If you are in any doubt at all about your batteries bring them into a local vape store and the staff there should be able to help you.