Most of the walks published on this website are in easily accessed non-mountainous countryside. However, a few of the walks do take you into the higher Ochils where the weather can be very changeable. What you take with you will depend both on the type of walk and the expected weather.
For a short easy walk not to far from civilisation you may need nothing more than suitable footwear. If you will not encounter any refreshment stops on the route, then it would be a good idea to take a few snacks and a bottle of water with you. If your route is not going to be straightforward to follow (most of the walks on this website are not waymarked trails) then you should have a suitable map or navigation device with you.
A hill walk however is a totally different proposition. In this case you need to be self sufficient to look after yourself for the full duration of the walk. You should also consider the situation that could arise if something goes wrong e.g. a minor slip causing you to be unable to walk back to your car. For such situations you need to consider whether the equipment you are carrying will enable you to survive out on the hillside (in possibly inclement weather) potentially overnight. For this situation a survival (bivvy) bag, extra warm clothing, additional food and a torch are essential.
A good summer hill walking equipment list is available from Mountaineering Scotland at http://www.mountaineering.scot/activities/hillwalking/getting-started/essential-kit. This list is pretty much the standard kit than every summer hillwalker should be carrying.
In the hills in winter when there is snow and/or ice on the ground you will need additional equipment. Please see http://www.mountaineering.scot/activities/mountaineering/winter-mountaineering/getting-started/clothing for information on winter equipment.
If you are looking to buy new kit, it is worth doing a bit of research before spending large sums to ensure you get the right kit for the job. The following websites should help you.
https://www.ukhillwalking.com/gear/
https://www.livefortheoutdoors.com/reviews/
tgomagazine.co.uk/outdoor-gear-news/gear-guides/
Many of the larger outdoor stores also have “buyers guides” on their websites to help you know what to look for when buying kit.