There are a number of general guidelines. Starting with basics, women will need a sports bra. This is a non-negotiable piece of kit and it is vital that this be fitted properly since it is designed to give you comfortable support during exercise and to avoid strains and injury.
If you plan to run in sunny conditions in the summer or the winter, you will need to consider purchasing sunglasses in order to reduce glare and prevent squinting. Equally, you should buy a good quality sunblock to protect your skin from burning and for comfort while running.
The decision regarding running shoes is one of the most critical. There are clear aesthetic and fashion considerations but the key requirement is for adequate support and cushioning and correct fitting. Your running footwear is likely to be the most expensive piece of sporting paraphernalia you will buy and you need to take time to make the right decision.
Shopping for tops will probably involve style and colour issues and you will need to decide on a budget. With this established, some thought will need to be given to the occasions on which you plan to run and the types of tops which will enhance your enjoyment and performance. For example, a man planning light daytime runs in mild weather conditions may be best advised to buy loose fitting singlets or tank tops for maximum comfort. A woman planning long runs in cooler conditions may need long sleeved vests made from a high tech fabric that will wick moisture away from the skin and prevent clamminess on cold days. You may need a range of tops.
The same sort of debate will attend the choice of legwear. A man facing race conditions will want to consider shorts while a woman planning a moderate run on a mild day may prefer Capri tights, which are calf length and ideal for conditions which are "betwixt and between." Again, you may need a variety to accommodate different needs.
The main point can't be emphasised enough: you need to be comfortable in your purchases!