Until you get Lasers and assembly table to make gates, there isn't a good way to do what you want really. There are upgrades for the barrels. You could make it so the barrel itself auto outputs directly into the furnace. You have to add the 1st structural upgrade, the hopper upgrade, and then the hopper facade on the side you want it to output from. This would elminate pipes between the barrel and furnace entirely. I would suggest adding a barrel for charcoal so you can build up a buffer since you should make it faster than you use it. You also should not use the High temp furnace generator as it eats fuel way too fast It is the least efficient generator in the game. The normal furnace generator should run a tree farm just fine. If you need more power, you can add a 2nd one. An Ender IO stirling generator would work well too and can be upgraded with capacitors and is also a good choice. Even Thermal expansion generators would be good because you can upgrade a single one with augments. You can make them more efficient or get more power. They make 80 normally but can be upgraded to 160, 320, 640 RF/t if you need the power.
Here are 3 very early game options that don't use BC pipes since you can't keep them from overflowing without gates. i made all 3 of these in under 5 mintues so they are simple.
The 1st on the left uses the hopper upgrades so the barrels automatically output. You need to add the level 1 structural upgrade and 1 hopper upgrade to each barrel. Then plac the hopper facade on the bottom of each one. They do not need to be configured. Just set the furnace to blue on the top and orange on the left.
The middle one shows using a hopper instead of the hopper upgrades on the barrel. It also shows how you can use translocators as well. The furnace is set to blue on the top and orange on the right. A transloactor pulls charcoal from the bottom of the barrel and feed the generator and they can never overflow.
The 3rd is the simplest with the barrel having upgrades like the 1st setup i made and no buffer barrel for charcoal, I wouldn't recommend this though. It is better to make more charcoal than you need and fill a barrel so you can use it to help make power for more than just the tree farm.