Anyone can start a web site and ask for donations but who is going to take you seriously? I mean what credibility do you have? Now if you mean donating to charities then you would need to register as a charity. Of course you wouldn't have to but people don't just give away money to any old web site.
Receiving payments can be done via paypal or moneybookers or others although the above mentioned are the most popular. You simply sign up with paypal, assign a bank account to it and can either put a paypal payment link on your site, or a shopping cart or even just your email address. After you sign with paypal you should spend a week or three reading through their complicated instructions!
When you say safe, i assume you mean how you can receive payments safely well this isn't a problem if you use paypal or other online payment services as a user would be re-directed to a paypal secure site then redirected back to yours once a payment has been made.
You could also set up your own payment processor, like your own paypal, and i have such a script that you can have for nothing. I can either find the link to the web site i got it from or mail it to you whichever is easiest, but in this case you would need a secure server or ssl certificate which you would have to speak to your hosting company about. This basically gives you an https address.
How do you start a website? Well, here goes! It all depends what type of web site and what skills you have. A basic site could be created i one of many ways either using ready made templates and adapting them or using an editor, there are many free editors out there. If you need something a bit more involved, such as a sign in, user profile etc then you may want to either learn a language such as php or learn how to adapt an existing open source script, which is not so complex but a bit time consuming. Once your site has been created you then need to put it on the www. This can be done from a home server, (cheapest but some knowledge needed plus your pc must be on at all times and you would have to regularly back it up) or you could rent a dedicated server (expensive but good) or a vps (for as little as around 17 dollars per month, next best option to dedicated server but depending on usage may be best bet, however requires extensive knowledge of servers), or could be shared hosting which is by far the cheapest and simplest, or even free hosting providing you had optimal control over the free hosting and could easily upgrade. You would also need a domain name, which could be a free one (i.e from http://www.co.cc) next best thing to a paid one if you cannot afford one (but you can get domain names for as little as around 10 dollars) but the drawback is in the long run where you may want to sell your name.
Some free hosting companies automatically give your first name as part of theirs, as a subdomain but you can later add your own names.
You can get free hosting here: http://4fr3.com
Here are some reasons for doing so: It's free and carries no ads and does not require you to post to forums for privileges. It gives you six sub and six add on domains from one account all at 300mb space. You get a proffesional cpanel control panel for uploading and editing files, a free web builder and automatic installer for forums, blogs, ecommerce sites etc and 6 mysl databases. It's php hosting so allows you to run php scripts. Also, you can upgrade cheaply and easily if your site ever gets to a point where traffic is high and can no longer be handled by free hosting. Also, this gives you an excellent training ground for uploading, editting, installing and understanding web sites with no cost to you. Anything else you need to know you can mail me, i'm on the net 24/7!