If I'm connected to a Tier 3 ISP then does that mean whenever I access anything over the internet, the traffic has to go through a Tier 1 ISP because I've read that Tier 3 ISPs purchase their bandwidth either directly from Tier 1 or indirectly via Tier 2?
The conventional defintition of the "teirs" is.
- "Tier 1" does not buy any transit. Relies exclusively on settlement-free peering with all the other Tier 1 providers to obtain routes to the whole Internet.
- "Tier 2" uses a mixture of peering and transit connections.
- "Tier 3" relies entirely on transit connections.
(note: not all ISPs may fit into these definitions precisely, for example an ISP may be transit-free but rely on paid peering or an ISP may be transit free and be unable to reach the whole Internet*)
In practice "Tier 2" covers a massive range of networks from networks that sit in one room to networks that span the globe. Smaller "Tier 2" ISPs can and do buy transit from larger "Tier 2" ISPs. Larger "Tier 2" ISPs can and do have restritive peering policies making it difficult for small ISPs to peer with them.
Now think about the financial motivations of an ISP.
- Sending traffic downstream to a customer makes money.
- Sending traffic upstream to a provider costs money
- Sending traffic sideways to a peer also costs money but less money than sending it to a provider.
A large Tier 2 may deliver the vast majority of it's traffic over peering and customer connections using it's Tier 1 upstreams mostly as a "route of last resort".
With all that in mind lets address your question.
If I'm connected to a Tier 3 ISP then does that mean whenever I access anything over the internet, the traffic has to go through a Tier 1 ISP
No, the tier 3 ISP may pass the traffic to one of it's own customers or it may pass it upstream to a tier 2 who manages to route it over a peering link or even route it to a customer.
* See for example the HE/Cogent fiasco on IPv6.
Network/provider tiers are about who pays for transit and who doesn't. Tier 1 networks are so large that they don't pay for transit with any peer (they get paid by the others). Tier 2 networks pay for transit with some carriers and nothing with others (they get paid by smaller carriers and customers). Tier 3 pay for transit in general (they get paid by their customers alone).
When it comes to routing, there's little you can derive from your provider's tier, as ar_ stated.