Stay away from cotton and other similar products. Once cotton gets wet (i.e. sweat), it stays wet for a long time and can promote chafing and other uncomfortable experiences. Instead, look for hydrophobic materials such as synthetic (e.g. polypropylene) and natural (e.g. wool, merino wool, etc.). The retail market is flooded with these types of materials now. Typically, the more expensive fabric manufacturers have spent time and research on how the fabric feels, fits, and works. The cheaper manufacturers, will use their existing templates for other clothing and just change out the material. What this means, as an end consumer, is that you can get cheaper shirt, shorts, and socks but at what cost to your comfort and usability.
It is definitely a balancing act between getting something affordable versus getting the most expensive product out there. Consider the climate you are running in. Is it sunny and hot all day? You may actually consider running in long sleeves with built-in sun protection. Are you running in a heavily wooded environment? Think about bug protection (e.g. built-in bug protection into the clothing and wearing a hat).
One major downside to synthetic clothing is the stink and funk after a few runs. Cheaper synthetic clothing typically smells quicker and is harder to get rid of. More expensive clothing might have antimicrobial agents built-in, or an odor treatment of some sort (e.g. Ex Officio). Regardless of clothing features, change your clothes immediately following your workout. If you don't, you're promoting the opportunity for gross skin issues including acne. When you are just starting out, it is uncommon to have multiple pairs of workout clothes, but it is worth the extra investment.
A big issue I haven't mentioned too much about yet, is chafing. Chafing is caused by the friction of two surfaces rubbing together. I would say this is more common in folks just starting out, or those who are running long distances. Do not take this lightly, or you can become uncomfortable pretty quickly. There are a variety of products out there on the market like, Body Glide, or you can use good ol' Vaseline. Feet, thighs, nipples, pretty much anywhere. Your body becomes desensitized over time, and you become more comfortable over time, but there is definitely a learning curve.
Well, if the post doesn't turn you on to running, I don't know what will. The good news is, I condensed many of my hard lessons learned into a short post. In truth, your first several episodes of running maybe uncomfortable. This could be due to a variety of reasons. Once you push past these barriers in the beginning, running becomes much more natural and comfortable. That being said, listen to your body and seek out medical advice if you are unsure.