All these services do is give you a name to point to your IP, and a means to tell them when your IP changes to allow them to update their records. That way your friends and you would only need to remember that name. The application they would provide (or your router, if it can be configured to) would simply tell them when your IP changes automatically so that you wouldn't need to do it yourself.
If you Google "setting up dynamic dns" you will get a number of step by step directions, but unfortunately almost all of them are from a few years ago and seem to use DynDNS (which as I stated above is unfortunately no longer free). The steps will be very similar with any of the other Dynamic DNS providers though. As for trusting them, I had no problems trusting DynDNS and they have served me well, and I can think of no reason these other services wouldn't be the same. I would suggest a free service since they are readily available, and as such you aren't trusting them with or risking any financial information. All they really have from you is your IP address, and they won't know or care what exactly you are using it for. Most people use Dynamic DNS to host services from their home desktop like remote desktop or FTP or other types of personal remote access. It's not like they would publish it publically so people can join your Minecraft server or anything like that. They won't open you up to hacking to any greater degree than you already are (security needs maintained on your side and is the same with a domain, subdomain, or raw IP).
Setting up your router is a slightly more advanced task, but if your router is capable, likely pretty easy. The exact steps would depend on what router you have though. If you happen to use the open source DD-WRT firmware that is compatible with a fair number of routers (this is what I use), it's as simple as just selecting your Dynamic DNS service provider, and entering your credentials with them (username, password, domain). The router will take care of the rest. Most routers would be similar.