One day in Venice on a budget-What to do?

Vaporetto from train station to St. Mark’s Square. See St. Mark’s. Perhaps vaporetto to Murano or Burano. From St. Mark’s, wander back to train station, checking out any place that looks interesting on the way.

Walk, eat gelato, and ride the vaporettos on the Grand Canal. Maybe see if you can get some free tickets to the rooftop terrace in the Rialto area.
Row Venice give small group rowing lessons on the back canals for about 100€ total, for 90 minutes.
Enjoy!

The cheapest way to see Venice is to walk. The path from the train station to the Rialto Bridge and onward to Piazza San Marco is a mob scene, but if you wander off that path you'll find little canals, hidden piazzas (campos) and the medieval atmosphere Venice is known for.

Vaporetto tickets are 9.50 euros each. A one-day pass costs 25 euros. There's a good 3-day deal your teenager would qualify for, but it won't be helpful for a one-day visit, because I think it costs about the same as the one-day pass.

The fee for an online entry reservation for San Marco is very low, 6 euros. If you can get one of those tickets before you arrive in Venice, that would be great. They always sell out before the day of visit, so that's something you'd need to take care of ahead of time. Lining up for entry (which might avoid that fee) can take 30 minutes or probably even longer, so paying for the online reservations seems worth it to me.

The basic entry doesn't include the gem-encrusted Pala d'Oro (an extra 6 euros) or the museum and view terrace (an extra 9 euros). The museum and terrace are very worthwhile if your budget will stretch that far. Those extra tickets can be purchased inside the church. I see that the website suggests a visit of 15-20 minutes just for the church itself. I'd consider that very short. It's not small, and it has fabulous mosaics all over. I bet most folks spend more time than that at San Marco.

Entry reservations for San Marco

Yeah, just walk around mostly. It is worth spending money on a vaporetto ride down the Grand Canal. Otherwise, in my opinion, the views and the scenery are the absolute best things to enjoy in Venice anyway.

The city itself is the main site to see. Put away the phone maps and just wander. Seriously! Take turns deciding whether to go left or right at all of the paths. When you’re tired, head to Suso for some delicious gelato, walk towards the Rialto bridge from there. I recommend going to San Marco later in the day when the day tourists have left. Have a wonderful time!

You can go quite budget, with some caveats. Transportation is walking, but as mentioned, the vaporetto ride at least through the Grand Canal is a great view and could be your only investment.
There are 122 water fountains to refill a water bottle....but toilets are not free. You may alternatively buy a coffee or small something and use the toilet in the restaurant.
Picnicking is pretty much not allowed, but you can buy a slice of pizza and eat it as you walk...same with gelato.

Restaurants in Venice are about the most expensive in Italy, and they are not--typically--the best, because for most of them the customer is a tourist rather than a repeat visitor. This is not the city to have your splurge meal if you are trying to save money.

The cheapest sit-down meal is likely to be pizza at a pizzeria. Usually there will be something else (likely more expensive) on the menu. On average, I've found a slice of take-away (or stand-and-eat) pizza from a small stand isn't as good as what you get at a decent pizzeria. But the slices can be perfectly satisfying, especially if you get something just removed from the pizza oven rather than a slice that has been sitting around and needs to be reheated. (Even those are perfectly edible, but it's a shame to settle for that in Italy.) One thing about those slices, though--they are very messy. You may be wearing pizza sauce for the rest of the day if you aren't very careful.

I'm an opportunistic tourist; when I get hungry I try to find a nearby place to get something I'm interested in eating. I have data service on my phone, so I can open up a Google map and check the ratings of the places nearby. There's no need to go to a place rated 3.5 or 3.9 if there are higher-rated places nearby. Normally the menu is posted outside so you can get a good idea of the price range. (Note the charge for bread and/or cover at the bottom of the menu, though.) I don't go inside places with no menu visible, though I suppose the bars selling cicchetti (little nibbles) probably only have the prices posted indoors.

Mini-supermarkets are a good source of cold drinks and snacks for less money than you'd probably spend elsewhere. I've bought decent single-servings tubs of tiramisu (less than 2 euros) more than once, but I don't know whether the stores always have forks and spoons available.

A beverage consumed standing up at a bar (OK for your teen--there will be water and sodas available) will be a great deal cheaper than the same thing consumed at a table. The same may be true for gelato, but I'm not sure about that.

I guess I like cheap pizza - because I've always found that the cheap take-away pizza slices (definitely for tourists) you get in Venice are very tasty, actually. There are places to sit down and eat outside - there are little parks and benches here and there though fewer on the main drag, Strada Nuova (the busy pedestrian street which is mostly a straight shot from the train station to Rialto). Many of the pizza places have little casual tables for quick dining as well.

Andrew H. Venice has some rather bizarre rules on where you can sit and eat in Venice. Only a few parks and not all benches can be used for eating....but enforcement may vary. Definitely not in St. Marc's Square and not on steps. I checked before my initial post or I would have mentioned that option.

Venice is an amazing place to explore - it sounds like the perfect spot for your family adventure! There are tons of great activities you can do on a budget there. If it's nice weather you can just walk around town and find awesome street food or cool hidden gems. If you get hungry grab a classic gelato and sit by the Grand Canal and people watch. Plus, the museums are cheaper during the week so you can take in some of the Venetian art.

I saw a couple of tourists in Venice being warned by locals (not policemen) that sitting where they were (on sturdy platforms of some sort? I don't remember) could get them in trouble. The tourists--not being readers of this forum--looked bemused but moved along.

With just one day you will be limited in how much you can accomplish so don’t try to schedule too much. The best thing about Venice is you can have the most fun by not spending any money to get into museums. Just walking around, getting lost, watching people, and looking in windows can be a great day in Venice. Others have suggested riding the vaporetto up the Grande Canal, and I would agree that it would be one thing you should pay for. St. Mark’s square is pretty much the tourist centre as besides the Bassilica there is the Doge’s Palace and the Corer Museum right there. With just one day you will not have time to visit them all so I would pick the one that is most interesting to you. I have been into all and I think I prefer the Doge’s Palace because it has a bit of everything. Grand rooms with spectacular art on the walls and ceilings, magnificent furnishings, intriguing history, and even a prison. There is always a gelato place to grab a snack, you can get a bottled Bellini at a store, and grab a quick pizza at a cafe on a canal. Stand shoulder to shoulder with other tourists on the Rialto bridge and watch all the traffic on the water, or duck into one of the many churches to gaze in awe at the architecture (many churches do charge a small fee to go in). You really can’t go wrong whatever you decide to do.

You all have convinced me to wait until we're there to hire a gondolier. We're going to miss the Terrace rooftop view and the Bovono(?) staircase because they are unavailable for the date we're in Venice. Thank you for all the suggestions!

CA_CO mom, if you're on a budget, consider a vaporetto rather than a gondola. A one-day pass will let you hop on and hop off at will, so you can wander the streets, markets, and areas as your whimsy takes you. That won't be possible with a gondola.

As others have said, just wandering around is a great way to enjoy the city. But try to get off the tourist trail or you may not see anything but the other tourists. Although I've just told you to get off the tourist trail, I will suggest a stroll through the Rialto Fish Market (it's free unless you end up buying some fish!) and work your way over to Squero San Trovaso to observe the gondola repair shop.

I've been to Venice five times and never been in a gondola. Why would I need to to enjoy such a beautiful, unique place?