Should I save my amulets and eggs and stuff until my pet is stronger? I know training gets more expensive but I read in a comment that eggs are useless after a certain point?

you should use eggs as you get them because they can't be used beyond tier 3; amulets i would save for later
If you are talking about the eggs some of the easy/medium challengers drop as 5th win loot, you cannot use those once your pet is T4. You can use boosters whenever, but you will level up fast if you use the training center and do BQs every day. Boosters just allow you to level faster but the higher the tier, the more stats you need to get to the next tier. People will save boosters later on to quickly go through the larger tiers.
@ jensen @ Wesker
Thank you! Late March is T3 already just because I train (without battling much) so it's good to know to grab all those Win eggs before I train them any more.
EDIT hi again Do you happen to know which easy + medium challengers drop eggs on Win ? I'm actually not totally sure which ones are "medium" (starting health below 1000 maybe?) but I just beat Erdoo 5 times and no egg. I want to make sure I get all the eggs before they're not useful :)

I would just battle them all and then you will get the eggs + the exp xD
Easy challengers: abox - willowisp on the default, angry arnold- possesed Esther on the secret tab
Mediums are: cursed irion- vulty on the default tab, blood bat - spa Blake on the secret tab.
You might have to unlock the secret ones
i have them marked off on my google sheet :p it should beee easy buttwing bomber eclipse pock saper riscors willowisp angry angry arnold coeli
medium cursed iron manticat mephysto muffin man oslyae tide blood bat lizard minion rictus
if you’re missing any of the secret challengers in general, there are a few topics linked in the battle chat that have their unlocks listed (the 1P loot google sheet, the unlockable opponent list, and i think 's strategy help directory has them too)
[edit] also re: your difficulty question; the only way to go off the tabs right now is to know that each difficulty is listed alphabetically :x so like aboxalypse through willowisp is easy, and then cursed irion is the first medium, etc. it's easier to reference one of the sheets instead.
thanks! yeah I keep having to track down your google sheet in other threads, there are so many different threads and spreadsheets that all these different stickied threads and other guides and whatever link to :P this guide is great
I actually do have blood bat unlocked idk where it came from. thank you!

oh, i meant i marked the egg ones off on my own google sheet that i use to keep track of who my pets have fought x) but yeah, we've got loads of really wonderful resources thanks to everyone, idk how i battled without them before this! it makes tracking down eggs and mod crystals so easy :D