The simplest way from the command line is to use either service or systemctl — depending on which OS you have on the Pi.
> systemctl status weewx
● weewx.service - LSB: weewx weather system
Loaded: loaded (/etc/init.d/weewx; generated; vendor preset: enabled)
Active: active (running) since Wed 2020-04-22 00:03:07 PDT; 18h ago
Docs: man:systemd-sysv-generator(8)
CGroup: /system.slice/weewx.service
└─3359 python /usr/bin/weewxd --daemon --pidfile=/var/run/weewx.pid /etc/weewx/weewx.conf
or
> service weewx status
If you want to see the process you can also us ps, that is roughly equivalent to the Windows task manager. Something like this will show you the weewx process:
> ps -efwww| grep weewx
root 3359 1 10 00:03 ? 01:59:47 python /usr/bin/weewxd --daemon --pidfile=/var/run/weewx.pid /etc/weewx/weewx.conf
Leave off the “| grep weewx” if you want to see everything running on the Pi.
Network traffic is trickier, as weewx generates very little traffic. If you are serving a web site, that traffic is going to be running through your web server, which could be all sorts of things like libhttpd, apache, nginx, or others. There are very few command line tools for network traffic, sar can do it if you have the systat package installed and enabled. There are lots of other tools like bmon, slurm, tcptrack, iftop, iptraf - each with a different set of plusses and minuses and features. Or if you’re a metrics geek like me you set up telegraf, influxdb, and feed it all into grafana dashboards — not something a normal person would do. :-)