Many people gripe and groan when they are faced with the step up to bronze-tier because cassiterite is a seemingly extinct resource. I recommend using black bronze, an alloy of copper, gold, and silver, the last two which are easier to get than tin if you have a bad seed. The ratio to get it is 16 copper : 2 gold : 2 silver
There are two ways I recommend to get gold and silver, Panning and Quartz mining. You can also get Omok gold pieces from the luxury trader, which can be smelted. EDIT: as of 1.22, gold omok pieces will no longer be sold by luxury traders.
1. Panning
The easiest method. Get your pan, a pannable block (Bony soil works best, with a MUCH higher droprate compared to the gravels and sands), and go at it. You can't get silver from sands/gravels, so use either bony soil or the second method for your silver.
2. Quartz mining (copper pick + copper hammer + prospecting pick needed)
While normally you would need to already have bronze to mine quartz, you can skip this with ore-blasting bombs. Get sulfur, saltpeter (found in some caves), and charcoal. Mix into 4:2:1 ratio to get blasting powder. Use blasting powder to make bombs, and then start looking for quartz deposits. Upon finding a quartz vein, use a prospecting pick to check for readings of gold and silver. Even if it's a low reading, start blowing up the quartz. Within, there is usually a few ores of gold and silver inside the quartz. Although it is rare, sometimes the gold and silver will be destroyed, so only blow enough up for a black bronze pick, then come back for the rest to mine manually.
Hope this helps!