How can I open channels with Eclair? I have set up a Bitcoin Core wallet with 0.3 bitcoins in it and an Eclair lightning node. The rpc userid and password is set plus all the other settings I am supposed to have, hopefully.

No problems with the Bitcoin wallet, and the Eclair software runs without showing errors. I managed to send a channel request and it filled up with 3100 channels and 889 nodes, so it's ok. Hopefully, I have sorted out my firewall to open port 9735 in both directions. I am using noip to fix my ip address.

But every time I attempt to open a channel, it sits for a couple of minutes then dies. No channels ever last. They show WAIT_FOR_ACCEPT_CHANNEL for a while, "Your balance" shows as N/A. The funding transaction's id shows N/A.
Then they die. Sometimes with "Insufficient funds", sometimes "Failed to open channel". 

I would really be very very happy if I could get a channel going, but I don't know how to debug it. Is there a trick? How do I select a node to connect to?
It seems you're having technical issues with the app. I would recommend to bring this issue to the devs on their forum: https://gitter.im/ACINQ/eclair. Also, please prepare to have all info such as the version of the app, your OS, mainnet or testnet.