Aside from the first win, are there any rewards to playing Tavern Brawl

There are a few pieces of exclusive content thusfar in Hearthstone. The first is a golden 'Gelbin Mekkatorque' card, that could be earned by using real money to make a purchase in Hearthstone while it was in the beta.

The golden version of this card is no longer obtainable. It was previously obtained by spending real money in the store during the game's beta test, prior to the full release of the game.

In addition, golden 'Elite Tauren Chieftain' was only earnable through purchasing a ticket to Blizzcon 2013, and thus also can't be earned any longer.

However, the normal version of each of these cards are still both able to be earned through crafting them.

Card backs will likely be exclusive to each season and event, such as the ones earned through levels in ranked on specific seasons, but thusfar, I haven't seen anything specific about this yet.

Game knowledge is very important in Arena, to play effectively you need to be familiar with the class you're playing and all the cards in the game. There is a large component of luck in Arena, both in the draft and in the actual games, but there is also a significant skill component.

The actual games are similar to the other modes, with the exception that you can't rely on the two cards per deck limit to guess which cards your opponent might have on hand. The decks are also very different compared to constructed and rely much more on strong individual cards than specific combos. But most of your skill from constructed play should transfer to Arena.

Drafting is the most difficult part for a new player, I'd strongly recommend to use a drafting guide like Trump's Arena Tier List to help until you're comfortable with it.

If you enjoy the Arena mode more than constructed play, just start playing it and don't worry. If you don't enjoy it, just ignore it. If you're good enough to get more than 4 wins regularly, Arena is also more efficient than buying packs.

In short, I'd recommend to play Arena once you feel reasonably comfortable with the general rules and tactics. If you win less than 3 times on average you might want to practice a bit more before playing more Arena. Then you're likely making too many mistakes in either drafting or playing.