Q:

Does bonding 3 1G NICs into one make a 3G NIC? 

For example, I have 3 NICs (1GB) and make a bonded NIC.  Do I get a 3GB NIC?  
Should I connect these 3 NICs into 1 switch?  If one NIC dies, do I still have the network between server and switch alive? 
Or should I connect each NIC to 3 different switches?  So if one NIC or one switch dies, I still have the network alive? 
Which mode should I use? 

A:

For example, I have 3 NICs(1GB) and make a bond NIC, Do I get a 3GB NIC? 

No.  You don't.   The specific details depend on the hash mode, but typical hash modes distribute individual connections to each member.   Even with a round-robin mode only possible between two directly connected Linux systems you will not get the full 3GB, probably more like 2GB. 

Should I connect

No idea.   We can't make a decision for you.   You need to decide if you need fault tolerance or better performance.   You need to figure out if your switches allow you to have a link aggregation group that spans switches, this is a pretty high end feature, that usually requires a stack setup. 
If you have no idea what you should be doing, perhaps start with a switch-independent mode for fault tolerance only and connect it to multiple switches.   This is a really simple setup and would require no config on your switches, but it wouldn't really give you much of a performance boost. 
Past that, just try some modes and run some performance tests that reflect your typical load and usage of this system in various configurations.   Pick the best one based on your results. 

A:

Yes, if a link fails, you will still have a connection. 
As for the bandwidth question, when you LAG these connections together, remember that each individual flow of data will only traverse the link that the flow starts on.  That one flow isn't shared among the other links. 
It's also worth pointing out that you would actually be better off having either 2 links or 4, due to the hashing of the bits. 
Explanation:
If the number of physical ports is not equal to power of 2 (not 2, 4 or 8), you will have unequal load-balance as well.  For example, pick a case,
Under port-channel X, we have 3 interfaces being bundled gi8/8, gi9/8 and gi9/12.  The load balance algorithm will need to use two bits from the hash. 
Two bits will give 4 different patterns (00, 01, 10, and 11) and the patterns are assigned to the physical port.  Therefore, the overall traffic is not 33% on each channel.  It will be 50% on one and 25% on the other two links.