Here are my favorite tips, but take it or leave it depending on what makes sense for you:
I agree with Lucy about choosing flight practicality/quality over the lowest possible price, especially if traveling with a baby - if you get stuck or massively delayed, you may end up spending on food/lodging
I use Going.com and Google flights to find deals on flights
Now is a good window since you don't have to pay for an extra seat for the baby, although 14 straight hours with a baby on your lap is a LOT. Not sure if this applies in your case but being able to breastfeed was a major bonus (I used a sling/wrap for privacy) for keeping a baby happy and quiet on a flight. Also, it's free!
Pack your own snacks, fruit, sandwiches for travel days b/c airport food is expensive!
To the extent you can with a baby, pack light and/or travel with a carry-on only. Airlines charge a lot for checking baggage now. There are a lot of videos on YouTube with good tips for packing carry-on only for overseas flights. I went to Europe last year with only this tote because I packed super carefully: https://the-foldie.com/products/the-foldie-4-0?variant=43022294810844
Since you have a baby, you should still be able to bring and check stroller, carseat etc. for free - check airline policies. Carseat bags make it WAY easier to bring a carseat to your destination, if you use one, and save on the high cost of renting one at your destination (if you plan on riding in a car).
Look into staying at Airbnbs or similar vacation rentals at your destination. It is sometimes cheaper per night than a hotel, and if it has a kitchen you can save a lot and even try local foods by buying simple items local markets and bakeries to eat for breakfast, snacks, etc. In the states where portions are larger, if you have leftovers from eating out, it's nice to be able to refrigerate them and heat them up later, especially with a baby when you might need to crash and eat at "home" instead of going out to dinner.
Having access to a washer and dryer is also helpful if you pack light to be able to wash clothes once during your trip. Especially because, as you know, babies aren't always good at keeping clothes clean.
Alternately, I found a nice and inexpensive hotel in Barcelona that had a kitchenette, so maybe look for something like that. I was able to get a small load of clothes washed and folded at a nearby laundromat for $10, which is still cheaper than checking baggage!
Maybe this sounds obvious, but cost check/compare accommodations based on proximity to where you want to spend most of your time and not just price per night. If a hotel or airbnb is the cheapest but far away from places you want to visit, you can lose money and time on transportation
Use public transportation instead of taxis or expensive tours
Search the heck out of ideas for "visit [your destination] on a budget" or "free attractions at [destination]"
Check out the city's travel/tourist passes that offer discounts on local attractions plus public transportation. They might be a good value, depending on how long you are staying and what you want to do.
I'm a fan of looking for beautiful public parks and historic libraries, which are often free to visit.