Airline messed up my upcoming connection, travel agent wants to charge fees for rebooking. What to do?

TL;dr – Speak to a local consumer rights organisation on this one before giving up hope or doing anything else.

Contrary to what the other answers have stated depending on a number of factors, particularly if you happen to live somewhere with stronger consumer rights or used a credit card under terms of such a jurisdiction, there may be some alternatives. Take all of this with a pinch of salt however as you’re not going to have an easy time winning any arguments here – everyone involved has a vested interest not to give any ground.

Before going any further it’s worth pointing out that the terms here do make references to a "Self-transfer Guarantee", so definitely double check the specifics of your booking because that would clear things up simply and conveniently for you straight off.

If our connection guarantee assistance services as outlined in this
section 8.2 (hereinafter referred to as the "Self-transfer
Guarantee"") is included in your booking this will be clearly stated
during the booking process and on your booking confirmation.

So there’s a few scenarios where you may have some recourse still.

Firstly you may be able to argue that the terms of the contract are that others have quoted are simply enforceable, for example in the UK there’s this in the Consumer Rights Act 2015:

"A notice is unfair if, contrary to the requirement of good faith, it causes a significant imbalance in the parties’ rights and obligations to the detriment of the consumer."

https://www.legislation.gov.uk/ukpga/2015/15/section/62

Note that I’m not sure this section actually applies to airlines or services, so if you want to go down this route definitely seek proper advice, e.g. from a consumer rights organisation (often free) or other professionals. Fundamentally as it stands though there’s a significant imbalance here in the terms others have quoted. They create perverse incentives for FlightNetworks because they stand to make additional revenue out of every schedule change and potentially can influence or understand the likelihood of that much better than you as an individual.

Secondly you may be able to argue that for instance FlightNetworks themselves cannot escape Regulation (EC) No 261/2004 – if airlines and their agents could escape that using small print you can bet they would’ve done so by now.

"(12) The trouble and inconvenience to passengers caused by cancellation of flights should also be reduced. This should be achieved by inducing carriers to inform passengers of cancellations before the scheduled time of departure and in addition to offer them reasonable re-routing, so that the passengers can make other arrangements. Air carriers should compensate passengers if they fail to do this, except when the cancellation occurs in extraordinary circumstances which could not have been avoided even if all reasonable measures had been taken."

https://eur-lex.europa.eu/legal-content/EN/TXT/HTML/?uri=CELEX:32004R0261&from=EN

(They might try to argue for example that they’re not an "authorised agent" here and so it doesn’t apply, but that’s probably a double edged sword because they’d also potentially be arguing they couldn’t actually sell you the tickets in the first place!)

These are just a few examples of why it’s not a clear cut scenario. In terms of how to proceed with this sort of approach you ultimately need proper advice, not just random people typing nonsense on the Internet. I’d brace yourself for them recommending you pay the fee initially (probably with some carefully crafted words about doing so under protest). Then in slower time they may suggest following up with some kind of action, either small claims court, or if you have something similar to S75 of the consumer credit act with your card issuer. (There were some notably consumer friendly judgments there in the past, see for example Mr M on page 6)

So, if I were in your position I wouldn’t do anything without at least giving a local consumer rights group in your jurisdiction a phone call and sound things out to see what they recommend. Maybe you are out of options, maybe you aren’t, but don’t let strangers on the Internet who don’t even know where you reside tell you that either way.

What can I do?

Not much.

FlightNetworks clearly states in the terms and conditions that you need to pay a rebooking fee in case the airline cancels.

Timetable changes and cancellation by airlines:

Your agreement with the
applicable airline may allow them to cancel or amend your bookings. We
will notify You of any changes once We are informed of them by the
airline.

If You wish to request a change or cancellation refund, as an
additional own service, We offer to handle the request on your behalf
if permitted by the conditions of the airline. For a list of our fees,
click here.

See https://ae-en.flightnetwork.com/terms-conditions#CUSTOMER_RIGHTS

It makes for an interesting read. FlightNetworks does NOT sell you a ticket, they "mediate" a contract between the airline and you on your behalf. This is a very messy concept since now both the airlines and FlightNetworks terms and conditions apply and FlightNetworks is not responsible for anything, if things go sideways.

I’m specifically buying cheap tickets because I have a very limited budget,

Sorry, but you get what you pay for. There is a reason why these tickets are cheaper. Here are some general guidelines

1. Always check prices with the airlines directly. In many cases the price difference is small or none at all. It’s MUCH safer and convenient to book directly with the airline, so make sure that the savings (if any) is really worth it.
2. Check online reviews on Trustpilot, Better Business Bureau, TripAdvisor, etc.
3. Find and READ the terms and conditions. Yes, that’s boring and tedious, but at least you will know what you get yourself into and can make an informed decision. To FlightNetwork’s credit they clearly spell out how they work. If you don’t like it, don’t book.

Not all budget 3rd party providers are bad, but there are some really bad ones out there and you need to assess your risk before you book. Don’t be pennywise and pound-foolish.

Buying from a third-party agent is not a mistake. They usefully different price-points with different restrictions and fee structure, giving customers more choice.

The main question is to determine whether the third-party sold you one ticket with two segments or actually two separate tickets back-to-back. This is normally disclosed to you before you buy the ticket, although this is not always clear to people who do not deal with such agents often. This conditions and risk of those 2 types of booking are very different:

• If what you bought is a single ticket, it is the responsibility of the agent to deliver you a ticket that makes sense together. Air Canada may have cancelled or rebooked a segment but if that makes the other one impossible, then the agent has to rebook that one or provide you with completely different set of flights to go from your departure to your destination. You can usually discuss options with customer service. This is something that happens fairly often when an airline changes their schedule.

• If what you bought is actually two separate tickets just timed to coincide then, you have to assume the fee to rebook one of the tickets if the other changes in a way that no longer works out. This is the risk assumed by choosing to book as two tickets which can be cheaper than the single-ticket option. In this case the agent will refund or rebook for you, at your request, and for the fee specified in the fare. Keep in mind that some fares are not refundable and canceling you will lose the entire price of that ticket.