What is the role of bitcoin wallets? I'm trying to get involved in bitcoins, but first it seems I need further clarification on the concepts involved in bitcoins.

First let me clarify what I know (at least what I believe to be correct): miners race to solve mathematical problems, and whoever (or whatever pool) manages to solve the most complex block at a given time is awarded the bitcoins to their wallet address.

Now, what I want to know is, what exactly makes a wallet wallet? Other than providing an address, what does it do? A pair of cryptographic keys make up a wallet, which I understand, is used to transact the bitcoins associated with an address; but is that all? Other than safeguarding the private key, is there anything else to mind about a wallet, i.e. does it record anything else?

I did not find anything explanatory aside from the key-pair mention on the Wikipedia article, and I do not have enough technical expertise to go through the bitcoin protocol to understand the concepts. Please help me with understanding wallets.
A wallet is merely a collection of pairs of private/public keys. Every normal Bitcoin address is just a hash of a public key. The corresponding private key is needed to sign transactions that spend coins from the address.

So wallet security really is all about protecting your private keys - making sure you have access to them, and nobody else does. There are several ways to do this. A basic client software manages a wallet file for you and uses the keys within to sign transactions.

There are more advanced address that correspond to multisig or other scripts.

Re mining - almost true, instead of the miner finding the most complex block winning, every miner finding a sufficiently difficult block is rewarded. What is considered "sufficiently difficult" changes periodically.