Tangos and Healing Salve and stock standard, and Bristleback needs these as he can take a lot of early damage against ranged heroes in particular.
Ring of Protection gives a great armor boost, and the two Iron Branch are the start of a wand. I sometimes get some Clarity potions instead of branches as your mana can be a problem early on.
Buying a Sage's Mask first thing gives you your Ring of Basilius and helps with your early game mana issues. The order of the next three items, finishing off your wand, Power treads and Stout Shield all depend on whats happening in game. I usually skip the Shield if I can as I don't always build it into a Vanguard.
Enemy Magic? OR Enemy Physical?
These two option depend on the heroes selected by your opponents. If the enemy have more magic-damage type heroes or lots of stuns/silences then get a Hood of Defiance that builds into a Pipe of Insight. You can also get the all important Black King Bar to help out. Linken's Sphere is also another option but having the first two should be enough.
If the enemy has more physical-damage heroes then I go down the armor boosting path (my favourite!). Getting Mekansm depends on the supports on your team, often they will get this but I am more than happy to as its gives great armor and health regen, though does drain your mana. Getting a Mek often means I will get another item to help my mana regeneration. Assault Cuirass is great on Bristleback for obvious reasons. I'm not sold on the Vladmir's Offering but unsure of a good, cheapish alternative if someone else is getting Mekansm. Often I will go straight for the Assault Cuirass.
Basically go with what you need (it may be a mix of both).
These are viable option depending on whats happening in your game.
Drum of Endurance is a handy cheap option that I am warming to.
Heart of Tarrasque is awesome but the Reaver you need to build it is a long wait till you save up the gold.
Abyssal Blade is great for damage/ganking, and you could just get the Skull Basher instead of turning it into an Abyssal.
Shivas Guard is a great mix between Armour and mana problem AND has a great slow.
Gem of True Sight is great on Bristleback once he has his tank items.
The Radiance I tend to only go if I need to do EVERYTHING on my team.
Shadow Blade only if I'm sick of dying when my team is getting smashed.
Vanguard tends to be only useful REALLY early and scales poorly.
Orb of Venom is okay but takes a slot and does what you should be doing with you 'q' skill.
Blade Mail is not effective on Bristleback due to his damage reduction.
Eye of Skadi looks good but for the price you are paying there are so many better options.
Medallion of Courage is useful 1v1, but as this is a tank role the negative armour it gives you when active is not good if multiple enemies are focused on you.
Viscous Nasal Goo (Q)
Viscous Nasal Goo is a snot rocket that allows Bristleback to effectively gank, however it become much more effective once he has his ultimate ability.
Quill Spray (W)
Quill Spray is AOE skill that when stacked gives huge damage. While it is handy as well to secure last hits, at early levels the damage is very low and can be hard to judge.
Bristleback is what allows him to be a great tank, reducing damage when his back is turned. Positioning is crucial for this skill (obviously) and you should find yourself constantly trying to have enemies hit your back for that free Quill Spray release.
Warpath turns Bristleback into an angry, angry trucker. Spamming your Goo and Quills gives him huge speed and right click damage, meaning opponenents have to TRY to take you out early in a teamfight or suffer the massive damage you can output.
My Skill Sequence
Again I tend to change this depending on whats happening in the game. People often like to put their third point into Quills before their second point into Bristleback, but I prefer the more defensive option. If I think I can get an early kill I will go Nasal Goo earlier than Level 4.
Bristleback is often used solo lane but in pubs chances are someone will lane with you. I prefer a ranged support that can harass the enemy heroes, and I can protect when a gank comes. My favourite laning partners include Skywrath Mage, Dazzle, Omniknight and Witch Doctor to name a few.
Not too many but the main ones are Axe and Legion Commander due to their abilities that force a face to face fight, which doesn't allow Bristleback to use his 'E' skill.