Currently training my Rreign to level 10. What are the best items for a tier 1 pet? Secondly, how do you actually gain XP for your pet's tier? The official guide says you increase your tier rank by just training, but it barely makes a difference.

T1 is really fast to get out of, so people just get any random set and stick with it until they get to higher tiers where there's more options to play with.

If you need a set recommendation:

If you want to make your own set, a typical set at low levels consists of 2 "infinite" use weapons (aim for at least 9-10 icons total at T1) and fill with either healing or single use weapons. This set is 2 healer items and 4 single use "bombs", but you can mix and match by browsing subetalodge and filtering out the T1 list. You can use this up to T3 when you get access to actual useful items.

A more defensive pair of infinite weapons are:

These have much lower damage compared to the Fire Blade and Darkmatter Sai, but gives you 16-18 Healing per turn. Keep in mind there are certain fights that do not allow you to heal at all, so it could be preferred to get both sets and change weapons for that situation.

Outside of battle quests, the best method to tier up at the lower levels is to defeat Easy opponents up to 10 times to get the max EXP from the fights. Not only EXP, but defeating certain opponents a number of times gives you eggs, which is a booster that you can use for pets below T4.

The bonuses you get from defeating opponents up to 10 times is per PET, so you can also use another pet to get another set of bonus items (e.g. more eggs to use for your main battle pet).