I think this is probably best split into two questions, since they’re very unrelated.
For hiding your IP address, the main question is: who do you want to hide it from?
I’m not sure what a “personal private server” might mean in this context, except perhaps setting up a server at home on your own network. There is nothing inherently unsafe about this, although if you expose it directly to the Internet you will need to become a little versed in standard systems administration security practices. It won’t have anything to do with hiding your IP address, though. It’s not the best for running an eCommerce website, though, because home connections aren’t designed for that (and your ISP probably forbids it without a business account).
A Virtual Private Server isn’t generally a privacy thing. It’s a server-class virtual machine that you rent in someone else’s datacenter. You might want one of these to run your own web site, your own remote backups, or even a Minecraft server. So, this might be what you want for running your own eCommerce site. This would give you some degree of privacy in that anyone using that service would not also get the IP address of your home connection, but wouldn’t make your home connection more secure.
A Virtual Private Network — VPN — is a setup where networking is tunneled through a secure channel to reach a shared network somewhere. Often, this is used in corporate setups so that you can access work systems from home or a coffee shop in a way that your company’s IT department can reasonably trust. Basically this “bridges” you in to the work network just as if you’re plugged in to a ethernet jack on the wall in the main office.
This doesn’t inherently give you privacy, but recently it’s become popular to use VPNs as privacy shields. When you are browsing the web normally from home, your internet provider can see all of your traffic. Encryption (https) means that they can’t see what you are doing, but they can see who you are talking to — for example, if you are visiting sites with certain medical information, or which have certain political affiliations, or so on. In the US, many internet providers explicitly track and sell this information about your internet usage.
Additionally, when you are browsing in this way, sites you visit can see where you are coming from. For example, as an admin on this site, I can see precisely where you last visited here from. We’re not going to abuse that here, but lots of places on the internet are not so good. At the very least, they sell this information to advertising companies who use it to link things together. This is why if someone at your house does a web search for “funny socks”, within hours you’ll have “funny socks” advertisements showing up everywhere you look.
A VPN provides a secure tunnel to a different provider, and so all your own ISP can see is that you are connecting to that VPN provider. They can’t see where you go after that, because the outgoing traffic appears to come from the VPN provider. And, sites you visit also only see the connection coming from the VPN provider. So, that provides some level of protection.
However, the VPN provider can still see where you’re coming from and where you are going. Again, https keeps your actual data secure, but not the “metadata” about what you connected to and where. Some providers promise that they do not log this and make other privacy pledges. At the very least, this means advertisers are stymied (unless the VPN provider sells you out — you have to have one you trust), and it means that casual inspection by site administrators doesn’t reveal where you live. However, VPN providers are still subject to laws, and you should assume that your connections can be traced by the authorities of the country where you live and the country where the VPN operator is based, at minimum.
You could set up your own VPN via Amazon EC2, but this is highly unlikely to give you significant privacy beyond what a commercial VPN would, and will be a lot of work, and could end up costing more. (VPNs for Amazon EC2 are mostly for the purporses of connecting servers running in Amazon’s AWS cloud into your own datacenter, not for individual privacy.)
If you need a greater level of privacy, you should look at using the Tor Browser, which uses a model called “onion routing” to make it very hard to track your actual origin, even for interested nation-state adversaries. It’s probably overkill, but I personally use this whenever I’m looking up anything medically sensitive.
But all of that said, if the eCommerce websites are your main goal, the best thing to do is simply find a hosting provider who offers such a thing as a package. Unless you are already equipped to deal with credit cards, you probably want someone who offers that as a service as well, because the security and rules around that are nightmarish. There are lots of companies that offer this — but it’s kind of outside of the scope of Fedora.
Hope this helps!