Personally, if I were just starting something (assuming non-aws), I would use a free pinging service or write a quick cron to do it. In the event that it did not come back healthy, I would either send myself an email or an sms - the health check can also report error counts or use storage to track number of restarts per unit of time. Logging would just be some basic logging rotation and stdout piped to that file. Basic metrics could be as simple as a metrics table that you can query with SQL. When you eventually need fancy graphs (maybe a bit later), there are lots of options.

I'm not overly familiar with aws services, but I'd be very surprised they don't give a basic good-enough solution to this with cloudwatch, which is even less work than outlined above.

Depends on your stack and requirements (do you want to know about errors ASAP, or is a 2-5 minute delay ok?), but I personally love NewRelic because of how easy it is to set up (and the number of features that it has).

If you want tools that you can manage yourself, then a combination of StatsD + Grafana for metrics, and Sentry for errors. For logs, Graylog if you want to set up something complicated but powerful, and syslog-ng if you just want to dump the logs somewhere (so you can simply grep them).

Recently I tried logz.io In free plan you have 3 days retention, 2gb of logs daily, automatic notifications (insights - your syslog is parsed and common errors are recognized). And they have one click set up of visualisations in kibana.