So my questions are:
1) what can I use to clean the table that is strong enough to clean but won't destroy the wood (or "wood product", see below)
2) do you have any suggestions for how to improve the appearance of the scratched finish? After cleaning, could I just find a matching stain to go over the scratches with? Or do I need to sand the entire surface and start from scratch? Any recommendations on what I should try and/or specific product recs would be greatly appreciated!

I'm going to try to add a couple of pictures and the link to the product site/info. It's from Pottery Barn Kids, so while it seems sturdy, I know (from a past experience, unfortunately!) that their stuff isn't the best quality to start with. Which is why I bought an inexpensive used one! Plus, in a couple of months, I'm sure my kids will have their own scratches to add, so I'm not toooo concerned with making it look perfect, just spiffed up a bit.

Thanks so much for any input you might have!
Jen

Some info snipped from the product website (the specific wood/veneer type is not stated, but it is pretty soft whatever it is):
- Masterfully crafted from solid hardwood, hardwood veneers and MDF, an engineered wood that lends exceptional strength and ensures that the table and carts will endure over time.
- All wood is kiln dried for added strength and lasting beauty.
- The use of veneers results in high-quality furniture with flawless surfaces and consistent color tones.
- Our finishes were developed exclusively for Pottery Barn Kids.
- Our Espresso and Honey finishes are applied in layers for exceptional depth of color. All finishes are sealed with a final coat of lacquer for extra durability.

Lacquer is a disolving finish. That means any application of an additional coat of lacquer partially disolves the top layer of the existing finish. So, I'd use a finish sander(1/4 or 1/2 sheet sander---NOT a random orbit sander-ROS) and 150 grit paper to sand out as many of the scratches as possible. However, if there is only one coat of lacquer, you will be into the finish quickly, even with a finish sander.

Use a rag dampened with lacquer thinner now and then before and during sanding. That will disolve the top layer and make sanding faster, even with the 20 minute waits.

Many of the scratches are through the lacquer and into the 'finish' they mention. Those 'finishes' are a recipe of stain/finish/additives that are propriatary to Pottery Barn. Without the actual recipe, matchig the color is virtually inpossible.

So, most of what you will accomplish is just going to be modifting the present look.

re: the MDF comment - Gotta love those marketing departments!