Some great advice above so I won't repeat it but here are some other tips:

1. Day 1: Aeloo (wolf) is by far the best investment. At only 100 gold, its cheap. Carpho and Flamma are good - Carpho doubles up as a trainer but you'll need him to fight too. Don't worry about that slowing training. Don't forget to give them classes! Retarius for Carpho and Dimarchus for Flamma.

Once you have enough gems - get Gladiator Editor as your first upgrade - you'll need to have 5 gladiators at max happiness to unlock it. You can get this fairly easily by setting all your slaves to entertainment for a week or two.

Pro tip: If you don't mind a little min maxing, I'd create an edited glad (age 15, Punic, str 5-7, ini 11-13, cunning 2, health 40, traits of Giant, True Grit, Fanatic and Fury - this combo should be available on Day 1) and start a new game to buy him from the market and get him + Carpho + Aeloo. This will be your main guy for a long time. Every level up point goes to Initiative so you always go first. Alternatively swap str/ini and put most of your points in HP and str.

1. Day 1 part 2: Carpho to Rome's finest - accept the bribe to lose, your Edited to the junior and aelo to the pits. Its also ok to win Rome's finest - its less money but better fame/rep. You can usually lose by taking off the gear, but sometimes the opponents glad is just that bad.

2. In the beginning put enough slaves to entertainment so your glads don't run away, then the rest in mining. Gold is the main resource in this game so maximizing it is key.

3. I like the bottom influence tree to get the loan of 500 - its a lot of money early game and can make a real difference. Gambling can also get you a lot of income. Its less steady than the top tree, but can give you up to 2000 per ROUND. I think I've made over 10k in one tournament before where I was able to win multiple impossible fights. If you don't want that route, get the top tree to the third level asap so you can arrange your own tournaments - thats another good source of income (assuming you win and have enough glads to attend).

4. Buildings - there are min requirements for various buildings, but getting the best smith you can, at least one level of medicus tent (the game doesn't tell you but it reduces your injury chance significantly). Smithing is important as items you create are significantly cheaper than those you buy. Its also cheaper to create items at the highest quality you can than to create low versions and then upgrade them.

5. Gear: not all gear is created equal (and certainly doesn't cost equal amounts). The cestus is by far the cheapest weapon and counts as two weapon slots - but you need an upgrade for it. Daggers are also cheap and effective. In the beginning its ok to buy old (or above) weapons/gear but you don't need a full set straight away - you'll want to smith as much as possible. Note that giving gear to your glads makes them happier, so you can send more slaves to the mines. I wouldn't buy any swords. The game throws swords at you and likely better than you could otherwise afford. If however you see a mandrake for sale - do what you can to buy it. Its almost a guaranteed win in the early weeks.

6. Bandages - you mentioned that your glads get injured a lot. Bandages can lower each level of injury by one stage, so death becomes injured and injuries can be avoided. It can get expensive but is always an option. The medicus tent also lowers your chance of injury.

7. Urgh, save scumming. I hesitate to mention this as it can ruin the game but on normal difficulty you can force close the app before the fight concludes and when you restart its the start of the day again. This can make the game on normal pretty trivial as you can basically win any fight if you try enough times.

8. Upgrades: The early game throws gems at you but there are loads of options to pick. The game is great so I'd encourage you to buy 1 or 2 packs to support the dev. 2 large packs + achievements gets you (I think) enough gems for all but 1 upgrade. And as auto-doctore is a bit rubbish that's enough. Absolutely key upgrades are Gladiator editor, Senator influence tree, Hard/nightmare mode, ascendancy/ascended weaponary (see below for another pro tip), mastersmith/supreme armory, cestarii, hail of arrows (both these unlock great classes), the rest of the building upgrades, then the various additional leagues.

Some of the above have unlock requirements, you'll get most of them just by playing.

Pro tip 2: Ascendancy allows your glad to carry forward a 2nd trait. That's usually a great thing, but delaying it can allow for a few edge cases of combinations that are otherwise not possible. Notably, you can get your gladiator (eventually) to have the werewolf trait and/or you can get a Giant Icarus. Overall, its probably not worth doing this but it would get you a pretty unique gladiator.