To share supplied, do I need to connect each settlement to every other settlement or can I have one large circle connecting everything?

No. As long as any two settlements are joined by one or more supply lines, they can share resources.

This makes no sense in text, so let me explain.

FROM Sanctuary TO the Castle: Sanctuary and Castle can share.

FROM Sanctuary TO Graygarden: Sanctuary, Castle, and Graygarden can now all share. This is because Castle and Graygarden are indirectly connected through Sanctuary.

For best results, it's much easier to manage supply lines if you use a central hub rather than create lines from settlement to settlement.

I use Hangman's Alley because it's map centered, and I'll either send supply lines to Hangman's Alley or when settlers arrive, send them out. When all the settlements have supply lines, it looks like spokes on a wheel, with HA as the center hub.

Edit: Found some helpful images:

http://rack.0.mshcdn.com/media/ZgkyMDE1LzExLzEyLzJiL0ZhbGxvdXRzdXBwLjI4NzUyLnBuZwpwCXRodW1iCTEyMDB4OTYwMD4/8dea3266/316/Fallout-supply-lines-640x360.png

http://cloud.attackofthefanboy.com/wp-content/uploads/2015/11/Fallout-4-Supply-Lines.jpg

Though, I shouldn't say "wrong way", because the paths will work, but it's inefficient and harder to manage.

I just have a chain of connections that starts in Sanctuary, and connects them all in a giant circle that connects again in Sanctuary.

And.. I actually found it easier to manage the supply lines when I did the chained circle as opposed to the central hub (which ended up being a total nightmare trying to manage). I tried the hub first, and found the chain was better.

Every settlement on the same network shares resources with every other settlement in that network. Note that they only share resources in crafting menus or workshop mode - you can't go into your workshop inventory in Sanctuary and pull out weapons or crops you stored in Starlight Drive-In, but in build mode those crops will be available for planting.

My first game, I did the chain as well, and found this much more difficult to manage. When I saw the Hangman's Alley video, my second game used it and it's been much easier to manage.

Of course, one has to get to HA as quickly as possible to take it over, but they're going to DC anyway...

I sent out 10 provisioners per settlement its a brahmin traffic jam on every road.

I like to mix between having a main settlement hub and having them daisy-chained. A smaller hub in each part of the map, daisy-chained with a few settlements in-between. I generally don't like making my provisioners travel long distances.

Same here.

In my experience, short routes to nearby settlements all connected in a chain is much easier to manage. I tend to run a few chains from Sanctuary Hills, and Hangman's Alley is certainly an important hub of sorts, but I have a few settlements that have more than one spoke. It depends on the map, the order in which I unlock settlements, and whatever works based on my settlement layout at the time. The main thing is that I always make sure I run the settler from the new settlement to connect to the network. That way, if anything goes wrong with the supply lines, I know which settler had the problem, and what their expected route would be (since it will always be to the nearest settlement connected to the network working backwards.)

I did a mix, first run through, with the main hub at Sanctuary...the problem with that was that the front gate/bridge at sanctuary turned into an absolute cluster dance with a mob of provisioners and pack Brahmin in the river and gate.

The additional problem is that some provisioning links can pass through 'hostile hotspots', where they pass close to raider/super mutant camps, or simply encounter cells, and if you are nearby you end up in a fight simply to protect your provisioner.

This run through, I've set up Red Rocket as a central administrative hub for the upper left section of the map, and intend to set up localised hubs as I expand, spiderwebbing out from each of those to nearby settlements, rather than daisy chaining all together in one long line. It's slower, but the benefits are that I can expand in an organised manner, rather than having to try to build up a lot of settlements at once.

This was sort of what I meant. At the moment, I have a few spokes to Sanctuary Hills, mostly from the early local settlements. Then there's one major line through Hangman's and Jamaica Plains. Jamaica Plains serves as a hub for the southern section of the map. I have another chain across the north and winding southeast, with County Crossing as a hub connecting eastern settlements.

Basically, as long as someone is organizing their supply lines in a way that makes sense to them, and avoids a few mistakes, they should be fine. Examples of mistakes would be creating a route that is too long/dangerous, or bottlenecking too many provisioners in a narrow chokepoint. I suppose one should avoid redundant supply lines, as well.