开始 - means to start (to do something)
开启 - means to open sth, to turn on some device.
例如:开始工作 - start working (开启工作 ×)
比赛开始了 - The match started. (比赛开启了 ×)
这将会是一个新的开始 - This will be a new beginning. (这将会是一个新的开启 ×)
开启电脑 - start up the computer. (开始电脑 ×)
开启开关 - turn on the switch. (开始开关 ×)
开启天窗 - open the skylight (开始天窗 ×)
Sometimes they can be interchangeable, but there is a subtle difference between their meanings:
开始新的纪元 - to start a new era
开启新的纪元 - to open up a new era
As for 统统 and 通通, they are basically the same. You can use whichever you like.