You'll want a weather station that will easily allow you access to the raw sensor data. There are two predominately popular choices. These are not the only choices but these two choices will allow you the most available options to expand with beyond just the standard use. The recommendations I will list are purely a personal opinion. Other people may have other opinions and other things that they use. There is certainly not a lack of options for you to consider but I can't exactly give you a comprehensive list of all the possibilities. I'm sure others will chime in with their opinions.
The two options I will recommend stem from two different ways of considering your weather station and what it looks like, what it does, pricing (your budget), and durability. Either of these choices have extensive support from this user community and from the manufactures.
You can go with a Fine Offset clone (from various resellers that brand seemingly the same stuff with some minor differences) or you can go with a Davis Instruments weather station. The Fine Offset clone brand to consider will depend on where you live as some of them are regional. These are Ecowitt, Ambient Weather, Froggit, Misol, Nevada Radio, and others. Ecowitt is the only world wide available one of these as it is the one that is a subsidiary rather than a reseller of Fine Offset. This choice represents a more cost conscious approach. It may not be as resilient as a Davis but you'll get good value and many sensor type options. The display console if that matters to you will be more modern. Another cons is that the distance between the sensors and the console will not be able to go as far as with Davis. With Davis you can go 1000 feet. With Fine Offset clones you'd be lucky to go too far past 100 feet....but some of the newer models go close to 300 feet. The Davis brand has two lines to choose from. There is the Davis Vantage Vue and the Davis Vantage Pro 2. The Vantage Vue in my opinion is a considerable expense that could be something to worry about if your humidity sensor craps out as they often do....the problem with this model is that you have to replace the entire outdoor unit and that is a costly expense. The Davis Vantage Pro 2 on the other hand has every part available at reasonable prices to be replaced by the owner. The Davis Vantage Pro 2 represents a more costly device that will last you a very long time. If you live in a coastal region the conformal coating should fair better than other models. Nothing is perfect in a harsh environment but it is among the best you can get. The Davis though does have the unfortunate downside that it looks like a system that has not changed in well over a couple of decades in design....with a black and white LCD display...again if that matters as to many people it doesn't and it is all about the sensors. Peak wind speed will be better displayed more rapidly on a Davis as it refreshes more often. Peak wind speed is not lost on Fine Offset...it is that depending on the sensor chosen will report less frequently and has a lower max speed capability. The other thing you'll have to contend with on a Davis is how to connect it to your network as there are many logger connectivity options; WeatherLink Live, WiFiLogger, Meteobridge NANO, or Weatherlink USB. These are all different with different pricing and capabilities.
Lastly to answer your question of how to send the data to your own website ....well either of the choices I mentioned will allow that with the proper configuration. You'll be able to expand and use some software like Cumulus MX, Meteobridge, Weather-Display, or WeeWX to basically send data via FTP to your website. You can design you own website to receive this data or you can pick from several website templates that are very well designed and simple to deploy.
So to recap the fist decision is Davis or Fine Offset clone (start by looking at Ecowitt). The second decision is what sensors and what logging option to use. The third decision is what software you want to use.
