This is just the essentials for getting started. As you develop in this hobby you may find other things you consider essential, but remember this is just for first-timers.
VG or Vegetable Glycerin — This is the main component of most e-liquids, a viscous liquid that produces clouds. You will need a large bottle of this.
PG or Propylene Glycol — This compound is used mainly to carry flavor and reduce viscosity. You will usually need far less of this, as you will also get PG from flavorings and, depending on choice, nicotine base. Some avoid PG altogether due to adverse effects.
Nicotine — Generally a 2% to 10% dilution (20 to 100 mg/ml) suspended in either PG or VG. PG based nicotine is easier to work with. If you want to avoid PG and aim for max VG, go for VG based nicotine. There is also the option of nicotine salts to consider, but you may want to start with regular freebase before you explore this route.
It is usually best to order these ingredients separately, but if you want a small budget investment just to try your hands at DIY, you can order a premixed combination. A common example is a 70/30 VG/PG ratio with 3 mg/ml nicotine strength. This means you only need to worry about the flavoring. It does however limit your options, the ratio will be thrown off and the nicotine strength will be reduced by the flavoring.
The quality of the ingredients has a big impact on the experience. Both PG and VG need to be pharmaceutical grade, so make sure the bottle reads USP, EP, BP, or your regional equivalent. High quality nicotine is essential, ordering from a reliable vendor in your part of the world can make all the difference.
Flavors — This topic requires a bit of research. Picking a handful of flavors randomly never leads to any success stories. The usual recommendation is to find highly rated recipes that sound appealing and order those flavors. Another route is to find the most popular and versatile flavors and browse reviews to figure out how to use them. No matter how you go about it, putting in some effort before you order will save you a lot of grief later on. If you don't care to learn how to mix and only want to save money, there is also the option of one-shots, premixed recipes.
Bottles — Starting out, you'll want a handful of 10 ml bottles for testing recipes and flavors. This is considered the smallest possible size for accuracy, without wasting too much on failed experiments. Some larger bottles such as 30/60/100 ml will come in handy when you've found recipes you enjoy. LDPE is the most convenient material as it is soft and easy to squeeze.
Measuring — A scale with 0.01 gram accuracy is necessary, and a 500 gram capacity will cover most of your mixing needs. Something in the $15-30 range will do, but avoid those with auto shut-off if possible. While you can measure by volume too, it is much more of a hassle and will save neither time nor money.
Pipettes — Small disposable plastic pipettes may come in handy as some flavor concentrates come in bottles without drop tips.
This is all you really need to get started, but start small. A liter of VG (or premix), 250-500 ml of PG, 100 ml of nicotine, 10-20 flavor concentrates and bottles, a sufficient scale and some pipettes is a good sized starter kit. You don't want to spend a fortune on your first order for two important reasons. First, you won't know for sure that you'll enjoy mixing until you've tried it. But more importantly, once you've tried some flavors and recipes, you'll have a far better idea of what to order next time. Then you can scale up your supplies according to your own needs and make much better decisions about what flavors you actually want. So start small and work your way up.
There is plenty of additional gear that may add comfort or convenience, such as funnels, condiment bottles, protective gear, an RDA for ease of sampling, the list goes on. But that is all optional and to a large degree depends on your personal needs and what your vendors have to offer.
When it comes to nicotine, local regulations may limit your options and there may be shipping restrictions. As for safety, even diluted nicotine needs to be respected. It's not very dangerous to a healthy adult with built up tolerance, some caution is enough for most people, but if you know you'll end up splashing it all over yourself, you might want to consider gloves or goggles. Be super careful with kids and pets though as even a few drops can be lethal to them.