All tickets come with associated fare rules which dictate if you can make changes or get refunds, and what fees and penalties apply in those cases.
Usually, the cheaper the flight, the less flexible it is: it can range from completely non refundable and non exchangeable to various levels of fees. Completely flexible tickets (which you can exchange to any other flight or get a refund on without any fee) are usually very expensive.
When changes are allowed, on top of any exchange fees, you will also usually pay the fare difference: they will price the new ticket (based on prices today, not at the time of the original booking), compute the difference with the price you initially paid, and charge the difference.
In most cases, as time goes by and flights fill up, prices go up, so it’s quite usual that you would have to pay more to change your flight.
Also, as usual, there may be substantial price differences depending on the actual flight and dates. Sometimes moving an another flight a little bit earlier or later can make a big difference.
You can check things yourself: try to book a ticket for the full new itinerary, and see what is the difference with the original amount you paid. Add any penalty fees you were quoted and you should get the same cost.
In some rare situations it can be more interesting to actually cancel the whole itinerary and rebook it, but this comes with quite a few risks, so if you are not extremely comfortable with it I would advise against it.
On top of what the airline charges (penalties/fees and fare difference), the travel agent may have their own fees, which can vary a lot.
And no, in this situation, the airline is very unlikely to allow you to change anything without going through the travel agent. In any case, it is most likely that the majority of the cost your travel agent quoted is from the airline, not the travel agent.
Still, in the future, unless you have a compelling reason to use a travel agent, it’s usually much better to book directly with the airline: whenever a problem creeps up, you then have a single company to talk to, rather than two who are likely to each say you have to talk to the other.
