Diego your question is too wide, but to give you a basis.
How To Do This?
basically there are several classes ready PHP that facilitate and much this process, look on the internet for web scraping, web crawlers, spider bot.
What do they do?
with these powerful classes you can call a function by passing the site and the id of the element that Voce wants to copy, for example:
$temperatura_atual = copiar_texto('www.google.com/site/previsao_tempo/sp/mooca/hoje', 'id_elemento_com_texto_da_provisao');
Easy, no? and now with this variable with the value, just save in the database. There is enough video on the internet.
Some libraries that do this kind of service:
https://stackoverflow.com/questions/4842371/what-php-web-crawler-libraries-are-available
Its north is: web scraping.
– Woss
Show. Thanks man!
– Diego Vieira