Many countries you travel to will require you to obtain a visa, depending on your nationality and the purpose of your visit. It is also possible to travel without a visa, depending again on the country and purpose of your visit. For instance, British citizens can travel to the USA for up to 90 days for tourism or business without a visa, but working would require a visa.
But China does require a visa for British citizens to visit. China does allow for online application for visas, so you can go to their official website, choose your nearest city, and follow the application process.
The thing you will need to choose carefully is the type of visa you apply for. China has many types of visas (e.g. tourist, family visit, work visas). Since you aren't married yet, you should probably choose the L visa (independent tourist).
You will need to deliver your passport and supporting documents to the Chinese embassy or consulate you have chosen using the instructions they give, and pay the visa fee. In a few days you will receive your passport and documents back. During the application process, you will choose whether you want them posted to you or if you will pick them up in person at the consulate.
Your visa will be a sticker placed on one of the blank pages in your passport.
After you receive your visa, check it carefully. Make sure all of the information is correct, and that its validity dates include the dates you intend to travel. If you find any errors, contact the consulate immediately, preferably by phone if possible.
If everything looks good on your visa, then you are ready to go!
Also, note that the UK Government publish country-specific information for British citizens travelling abroad. It is a good idea to review the information available for any country you are planning to visit.