This is the best question for a runner to ask, we have all types of important running gear which we need to use but my view is that the most important thing is the best shoes for each individual runner.
Once you have your own personal shoe which works for you, everything else follows on.
Explore online shoe reviews, this will give you some very good signposting towards a shoe for you.
Quality First: First of all, forget budgeting or buying cheap, the core of running ability is a good quality pair of shoes, this is a top priority.
Comfort is second: The more comfortable the FIT of the shoe the more likely it is that the shoe will perform with the individual mechanics of your body if something feels comfortable it is generally right for you. Look closely at the cushion system designed into the shoe, try the shoe on and see if the cushion factor is right for you.
Flexibility and sole grip/traction: Hold the shoe firmly, bend it in your hand and see how far the sole will let the shoe bend, can you double up the shoe by bending it and if yes does the shoe spring back to shape? Talk to the fitting shop and explain which type of running you are doing. Do you want a lot of flexibility or do you want a more rigid or balanced flexibility? I suggest you go for mid-way, a sole that gives some spring back when you bend the shoe. Look at the sole, which style of grip/traction is designed into it and does this work for your type of intended running, trail shoes have lugs for grip in muddy conditions but a road surface shoe does not.
Underpronating vrs Underpronating: every foot has subtle differences, talk to a shoe fitting shop, make them understand that you're going for FIT, they will guide your choice of shoe. Consider a gait analysis along with a 3D scan of your foot, this is available for you because the fact is that you want exactly the right shoe for you.
Minimal and Barefoot running shoe styles: Generally you're looking for a running shoe with a drop around 8mm between the cushioning from toe to heel but if you want to try the barefoot or minimalist style of running shoes you need to inform the fitting shop that you're looking for a zero drop shoe. This style of shoe goes for a natural strike to the ground with the ball of your foot as opposed to the heel striking first.
Neutral -Free motion: This style of shoe allows your foot to move freely in the way that it wants, there is debate about this shoe style due to pronation claimed issues such as the foot rolling and taking a lot of shock impact.
Think about a Neutral shoe when you make a choice, they are designed to maintain foot alignment because the design is focused on the amount of pronation by the foot.
Take into account stability: brand running shoes continue to advance shoe technology for giving stability through innovative tech which provides support to the foot. You can look for a nicely cushioned shoe, a slightly stiffer forefoot with firmness so you will get a more responsive feel when you're on your toes the forefoot will feel stabilized. Take a look at the Asics shoe line, especially the 2000 shoe model. If you like an emphasis on foot transition check out Hoka running shoes.
This is a lot to remember when you are choosing a pair of running shoes to suit your needs, remember to talk with staff in the shoe fitting shop, use the advice here to give them suggestions for shoes which you can try on for FIT.
Lastly: When you have your running shoes, relax, run in a style which suits you, your gait and enjoy your personally and carefully selected shoes, they are very important.