For log aggregation I use Elasticsearch + Logstash + Kibana. For statistics I use Influxdb + Telegraf + Grafana. For service alerts I use Consul + Consul Alerts. Notifications get sent to Pager Duty and/or Slack from one of Kibana, Grafana, or Consul.

The Grafana graphs are especially fun to watch and Kibana has a nice logentries-esque plugin for searching and tailing logs.

It's quite the system to setup but once it's running everything else (i.e. the rest of your app) is much more manageable.