Can't speak to the what a 3 year old would like aspect, but I would say everything is very accessible by walking. You can easily get to the bridges to the old city within a few blocks (and by not cutting through the Mirabell Gardens, avoid stairs). From there everything in the old city is very compact.

I'll say, with luggage, it was a bit of a pain to get to the hotel from the main train station due to the volume of people and bikes, but it's not really work taking a taxi or anything.

I'm not a huge fan of the city cards when traveling - they tend to be overpriced if you're not cramming everything possible into a day and we tend to be more relaxed travelers - but the Salzburg Card was so worth it. They sell them at the hotel (and room charge them). It's very reasonably priced compared to other cities and has the standard free admissions and transit access - with skip the line at major sites, which can be a great benefit in the high season, I'd guess especially so with a kid.