A lot of firewall products allow you to set up notifications by email when a particular event happens. The only thing I would say is that you may end up receving a huge number of alerts.
There are automated and massively distributed scanning netorks in China that constantly probe for vulnerabilities and the Russians have a fair amount of scanning activity too. The Chinese attempt is much more sophisticated in the way it's coordinated but the Russians are sneakier, launching scans from IP address ranges which have been falsely registered as belonging to legitimate and unsuspecting UK companies.
You could easily get an alert every few minutes. I've just had a look at one of my firewalls (with three external IP addresses) and there have been 445 connection attempts denied in the last hour. Another firewall with five external IP addresses reports 369. Basically, more than one event every ten seconds. It's going to be less than this if you have only a single IP address but it could still be significant.
Fingbox will alert you any time a new device joins your local network and this alert is sent to your mobile phone and email address. You can then block the device from your mobile phone so it does lock down the local network to an extent.
What it doesn't do is alert you if an external inbound connection is made to an existing local device. Having a decent router with an inbound traffic filter set to deny all (and not having exposed UPnP) helps mitigate against this.