You can look up videos on You Tube to leanr how to inspect those things yourself. Let the shop take a stab at it and then see if you can confirm their diagnosis with what you learn on YT.
A dealer told my friend that they needed tie rod ends. Tie rod ends get loose over time making the steering vague and causing the tires to wear unevenly.
Friend brought it to me and I was able to duplicate the play in the joint but the play was tiny. A decade or more later the vehicle has never had the tie rod replaced and the play has not worsened in any appreciable way. Friend does not drive alot of miles with this particular vehicle.
www.youtube.com/results?search_query=checking+tie+rods+and+ball+jointsThe first result is the channel "ChrisFix". He generally makes good videos.
With the age of your vehicle I would look for ruined rubber parts - V-belts, timing belts, serpentine belts (not sure what a Corolla has), shock absorbers perhaps, and CV axle boots.
My 20 year old Honda has wornout its front sway bar bushings three times ($12 DIY with NAPA parts) while its other suspension parts are just fine at 300K+ miles now. No axle boot problems. Not shock absorber replacements. I have noticed the upper balljoints need to be replaced b/c the rubber grease boots are rotted. Its worn out several sets of brake pads, one set of rear brakes, brake rotors replaced twice, rear drums machined once.
My other 20 year old domestic sedan has needed shocks and that's it. It has ~160K miles. Its worn out a couple sets of brakes before I bought it.
I think your Corolla will check out just fine as long as your local streets aren't like those of a third world country.