So you have decided on your requirements for your personal weather station and if you have decided to log the data for future reference or to upload it to the internet then you will need to have something to store all the data. You have several options availabler
With some exceptions all but the first scenario will require you to have a computer of some description. A lot of people press an old Windows laptop into service for data logging. If you have one or can get one cheaply then this can be an inexpensive method of logging your readings. I say Windows because most of the software you can use will run on Windows 7 onwards. Some will even run on Windows XP
If you are up for a challenge then the Raspberry Pi, which I use, can make a very good data logger. Be warned though it is not an easy task setting up a Pi or similar tiny computer unles you are into Linux and driving everything from the command line. That said, they are a very low cost and cheap to run data logger. Slightly more expensive are the tiny Windows 10 Bing edition PCs. Barely bigger than a lunch box and costing about £100 they will happily run Windows thereby opening up a wide range of wether softwre. There are other little computers running Linux such as the Banana, Orange and Beagle boards.
If you are a glutton for punishment then a number of routers can be reprogrammed to act as little data loggers and there are small weather station loggers such as the Meteo Hub. If you are sufficiently nerdy then I'm sure you can Google how to use them.
My latest weather station does not have a console but a small receiver and internet connector. It is easy to program it from a web browser on your home network and have it upload your weather readings to Wunderground. Being the person I am, I actually intercept the connection to log the data initially into my Pi then that uploads it to Wunderground, CWOP, WOW and this site here at Magull.org.uk thus enabling me to do a lot of data manipulation before publishing it on to the web.
Chosing the correct software to meet your needs is very important. Although it is possible down the road to change the software you are using it may mean that you lose all the historical records you have generated over the years. Most companies allow you to try before you buy if it is a commercial system. Open source on the otherhand