I have come across another piece that also does not have a hallmark, but is marked "sterling". Can you help lead me to the maker?

Thank you! I had read some people polish their silver with "silver wipes" purchased from Bed Bath and Beyond- are you familiar with this product at all? Do you think they would be sufficient or could they cause damage? The two pieces pictured in this post are a good example of the current shape the entire collection is in (the platter being one of the worst and the small bowl being one of the best- there are 15 serving pieces and a full flatware set). Would the platter be considered in poor condition?

I appreciate any and all thoughts on this, I really want to make my Grandmother proud!

On the first tray...

I don't know how often silver companies used the same pattern for DIFFERENT patterns, if you know what I mean. I have a large rectangular tray with the same pattern in the bottom but a different edge pattern. They might be the same pattern, just the larger trays had less elaborate edges, or it could be a different pattern. BUT...mine is marked "Wilcox, International Silver Co.". The pattern is Rochelle.

So you might have a different pattern, but I betcha it's International Silver.

I like either Wrights silver polish, or International...both are a pink soapy paste. I polish at the sink and use the polish as a wash...it suds up rum, rinse and repeat!
The top piece without any marking is an inexpensive (??) meaning not Gorham not Towle, silver plated tray. I have several very similar with the cast trim pieces. Treat it gently, the silver will wear away, likely leaving copper beneath. Looks to be 40's to 60's.
the little bowl marked "sterling" is again typical of inexpensive (!!!!) silver items. Rather run of the mill, not much design distinction, not marked with a maker's name....but it IS sterling!!
I make ( sometimes!) bags of silver cloth. Buy some on line and run them up on your sewing machine, leaving a flap to close. You can store several traps in one bag. Polish the silver before you store it or you are just adding the tarnish to other pieces.
let's see what else you have!!

How exactly would one "clean" silver? Warm water, a mild dish soap and a super soft rag? I'm sorry but I am an extreme beginner.

If you stand at a sink and use Wrights or international with a very wet sponge, it will clean as well as begin the polish. Or wash it with dish soap like you would a plate. Then polish. Either way the thing to remember is to get rid of the grease and grime before you try to polish. But if your silver has beenw rapped and in storage, chances are it not dirty, just tarnished.

I wouldn't take the time to "deep clean" silver. I mean really, it's not likely it has dried on mustard or cheese sauce. A quick wash like for dinner ware is enough. And if you have really fine antique silver...or any silver that you want to be really really careful about, don't use the pink stuff at all.....it's among the most abrasive and will eventually wear away your design and plating if it's a plated piece.
Curators and serious collectors recommend Blitz and Herman's. They work well if you don't have a really grungy piece but sometimes you have to do the stuff with a little jewelers rouge.
I am not a serious enough collector to worry too much...I don't have any Paul Revere pieces but do have a few things from that era. So I use the pink stuff but with a gentle hand>
Here's what the curators and restorers recommend. But he is touting his own product!
http://www.hermansilver.com/silver-polish-abrasion-ratings.htm

I'd never use TSP on silver, its formulated to strip walls before painting.