I was playing yesterday and today. Lately though, when I start the game I get a "connecting to destiny servers" message and eventually time out. I then have to close the game then restart it...then it logs me in.

D1 servers are up and running as normal. Any issues are on your end. Run a full internet diagnostic looking for issues. Look up any error codes you receive on Bungie.net or the Destiny app to bring up a help guide. Reset your router and then clear your system's cache to refresh your internet connection. If you use it no wifi as they are subject to interference but if you don't have the option to use a link cable use wifi boosters.