These notes from June 2016 document how I got a successful start with a Wemos D1 mini on Arduino (Mac).
I'm on Mac OS Sierra, so I need a driver for the USB serial bridge on the board, which is a CH340. The driver can be downloaded from this page:
https://blog.sengotta.net/signed-mac-os-driver-for-winchiphead-ch340-serial-bridge/
After unzipping, an installer can be executed, which unfortunately requires a restart of the computer.
The ESP8266 has provided libraries for the Arduino IDE. To install them, an additional repository URL has to be added to Arduino.
In the Arduino IDE, open Arduino > Settings and paste this URL in the input field Additional Boardmanager URLs:
http://arduino.esp8266.com/versions/2.2.0/package_esp8266com_index.json
Now open the menu item Tools > Boardmanager. Enter esp8266 in the search field and select the item esp8266 by ESP8266 Community.
Open the Tools > Board Menu and select WeMos D1 R2 & mini.
Connect your D1 mini to an USB port of your Mac.
In the Tools > Port menu you should see one entry like /dev/cu.wchusbserialfd120, which you select.
Open the EPS8266 specific blink example sketch via the menu File > Examples > ESP8266 > Blink.
Hit the Upload button in Arduino and watch the console. In case you see errors, try setting the upload speed to a lower value via the Tools > Upload Speed menu, then try Uploading again.
In the case of success, the blue LED on the D1 mini should start blinking fast during the upload. Then, when done, the actual program will be executed, which lets the LED blink in a slow and steady rhythm.
The next best thing is to try out some of the networking examples. It shouldn't take more than configuring your Wifi network's SSID and password in the Arduino sketch code to get a working connection.
Have fun!