In general, anything going through the proxy could be monitored, recorded, altered etc. by whoever is running the proxy.
There are different types of proxy:
A "web based proxy" sounds like a web page where you can enter the URL of another website, a bit like using a search engine, and it will fetch and display that website. I wouldn't use anything like this for entering any personal information, it might be of some limited use for working around a block on your IP or geographic location, or some problem at your ISP that is temporarily preventing access to a website. This kind of proxy may not work very well, depending on the complexity of the website you ask it to fetch. It would be limited to whatever website you entered into the box on the proxy's main page.
An old-fashioned HTTP proxy server would be something you would configure in your web browser settings. It would be more suitable than the web based proxy if you wanted everything to go via the proxy, it might work slightly better, although it might not fully hide your real IP address either.
Neither a web based proxy nor a HTTP proxy server would affect emails sent or collected by an email client, they are only for web traffic.
A better system would be a VPN, the security and privacy would still depend on whoever is operating the service, and perhaps which country the VPN servers are located in, which would be where your traffic comes out of the VPN.
I don't really have any experience of VPNs, a good free one would probably be an oxymoron. The trouble with operating a free proxy or VPN and allowing anyone to use it for anything would of course be that it would be used to do all kinds of bad things.
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Tor Browser might be suitable if you only want something for use with certain websites, I have not tried it.