This question is sound as it shows your making that first positive step towards exploring if running is right for you.
As a person new to running you should not overly invest at the start, invest just enough to get you started and gradually build up your running kit from this moment.Naturally you are looking to avoid any common mistakes made by people newly into running.
You really do not need to invest a lot to get started in running, if you have a "good" pair of standard running shoes already, you can even use them while you " find your feet" in your new choice of sport. If you do not yet have a pair of running shoes, you might read a reliable review for at least a suitable pair of running shoes, then choose a pair which you feel will work best for you and your budget. The qualities you will look for are, durability, comfort, protection, stability, wick-ability, cushion, traction/grip all of which result in over all bio-mechanical support.
However their are kit items which I suggest you invest in right from the start, breathable shorts, shirt and socks are a must so that you start your running hobby in clothes which are comfortably, functional and specific for runners. As a new runner you will work up a sweat and you will appreciate running clothes with breath-ability.
Clothing for running has come very far, the use of synthetic fibers is usual and these fibers are designed specifically with runners in mind, when you sweat, the moisture is wicked away from your body. Modern running clothing gives none chaffing performance for skin, many have anti bacterial qualities and come in lightweight comfort and tend to have built in reflective qualities to ensure your visible if running at night. Therefore I suggest any clothing which you do invest in should be a brand name known well by runners.
You may well be just like all of us, have a few cotton casual shirts, don't be tempted to wear these, cotton retains sweat, when wet it becomes heavy and uncomfortable and the friction of a wet cotton shirt on skin can result in painful skin rashes, the same problems can result from cotton socks and shorts.
You will have heard about running kit which just to name a few includes a GPS, cadence watch, heart monitors etc, you don't need this at the start, as you develop your interest in running and you give more focus and commitment you certainly can check advisory reviews so that you can add to your running kit. If you want to start of with monitoring your runs in terms of time and distance you can consider investing in a good fitness watch.
Now your ready for running and my last piece of advice is to consider putting together a nice running schedule for beginners so that you can enter into running and comfortably develop as a good all round runner for your enjoyment and physical health.