user11209: I am trying to understand Bitcoin at a more fundamental level. I'd like to know how a new transaction enters and propagates across the network? I've read many Bob and Alice type scenarios where it's said that "the transaction is broadcast to the network". If there are many nodes on the Bitcoin network, how does the wallet know which node to initially broadcast the transaction to?
EDIT
Just for clarification, I'm referring to the process of sending from the client wallet (e.g. Mycelium on Android) to Node 1, before it gets propagated to all other nodes. How does my wallet know which node to send to first, or does it just randomly pick one?

user13413: 
A node learns about a new transaction, either from a peer or of it's own creation. 
It announces to every one of it's peers (8+ typically) with an inventory message that it has learned about a transaction or block with a particular ID. 
If the connected peer doesn't already know about the announced transaction, it sends a getdata request back asking for the contents of the transaction. If it already knows about the transaction, no getdata is made in response.

Within a few seconds (less than 5s on average currently) every peer in the network has the relayed transaction. This is possible because of the mesh layout of the network, every node connects to every other node through some degree of separation. The design makes it incredibly difficult to determine if a node created a transaction or is just relaying it, this is by design.

user8717: I'd suggest you read this blog post about using the raw Bitcoin protocol: http://www.righto.com/2014/02/bitcoins-hard-way-using-raw-bitcoin.html
To answer your question, you should specifically read the "How to find peers" section of that post. Specifically this part:

There's a chicken-and-egg problem, though, of how to find the first
  peer. Bitcoin clients solve this problem with several methods. Several
  reliable peers are registered in DNS under the name bitseed.xf2.org.
  By doing a nslookup, a client gets the IP addresses of these peers,
  and hopefully one of them will work. If that doesn't work, a seed list
  of peers is hardcoded into the client.

and

The hardcoded peer list in the Bitcoin client is in chainparams.cpp in
  the array pnseed. For more information on finding Bitcoin peers, see
  How Bitcoin clients find each other or Satoshi client node discovery.


11209: <p>I am trying to understand Bitcoin at a more fundamental level. I'd like to know how a new transaction enters and propagates across the network? I've read many Bob and Alice type scenarios where it's said that "the transaction is broadcast to the network". If there are many nodes on the Bitcoin network, how does the wallet know which node to initially broadcast the transaction to?</p>

<p><strong>EDIT</strong></p>

<p>Just for clarification, I'm referring to the process of sending from the client wallet (e.g. Mycelium on Android) to Node 1, before it gets propagated to all other nodes. How does my wallet know which node to send to first, or does it just randomly pick one?</p>
 13413: <ul>
<li>A node learns about a new transaction, either from a peer or of it's own creation. </li>
<li>It announces to every one of it's peers (8+ typically) with an <code>inv</code>entory message that it has learned about a transaction or block with a particular ID. </li>
<li>If the connected peer doesn't already know about the announced transaction, it sends a <code>getdata</code> request back asking for the contents of the transaction. If it already knows about the transaction, no <code>getdata</code> is made in response.</li>
</ul>

<p>Within a few seconds (less than 5s on average currently) every peer in the network has the relayed transaction. This is possible because of the mesh layout of the network, every node connects to every other node through some degree of separation. The design makes it incredibly difficult to determine if a node created a transaction or is just relaying it, this is by design.</p>
 8717: <p>I'd suggest you read this blog post about using the raw Bitcoin protocol: <a href="http://www.righto.com/2014/02/bitcoins-hard-way-using-raw-bitcoin.html" rel="nofollow noreferrer">http://www.righto.com/2014/02/bitcoins-hard-way-using-raw-bitcoin.html</a></p>

<p>To answer your question, you should specifically read the "How to find peers" section of that post. Specifically this part:</p>

<blockquote>
  <p>There's a chicken-and-egg problem, though, of how to find the first
  peer. Bitcoin clients solve this problem with several methods. Several
  reliable peers are registered in DNS under the name bitseed.xf2.org.
  By doing a nslookup, a client gets the IP addresses of these peers,
  and hopefully one of them will work. If that doesn't work, a seed list
  of peers is hardcoded into the client.</p>
</blockquote>

<p>and</p>

<blockquote>
  <p>The hardcoded peer list in the Bitcoin client is in <code>chainparams.cpp</code> in
  the array pnseed. For more information on finding Bitcoin peers, see
  <a href="https://bitcoin.stackexchange.com/questions/3536/how-bitcoin-clients-find-each-other">How Bitcoin clients find each other</a> or <a href="https://en.bitcoin.it/wiki/Satoshi_Client_Node_Discovery" rel="nofollow noreferrer">Satoshi client node discovery</a>.</p>
</blockquote>
