We waited to start full efforts till we had the first $3000 USD saved up. I bought my domain name for the next 10 years, my main web coder is hosting it on his server, we aren't near Alpha yet, we are hoping early next year or middle of next year for it.
The main starting cost will vary based on what YOU can do as the owner or financial provider. I'm extremely lucky to have worked on the breeding team for Aywas and been an artist myself for a long time. Me and my Lairmate from Awas are heading the art side with just the two of us currently to belt out 4 pet species at least for Alpha and about 5-10 colorations per. then we are going to work on an item compendium of 100-200 items to start, and we'll be looking to hire someone to code games for us or implement in scoring systems/rewards systems to pre-built game codes that can be purchased for commercial use.
Out of my $3K so far I've only used about $200, because for my lair mate and I this is our Passion project and just a dream of ours we want to pursue while maintaining our current jobs. The coder wants to be paid for 10 hours of work at $90 an hour, but he's on standby and currently just tells me what he needs me to have ready to make pages functional while he runs the petsites he bought about 5 years back as that's his main workload besides freelance coding. My Roommate and long term RP partner is also going to help with Lore Writing with me so that also cut a huge cost.
Your flexibility, initial staff and such will determine how much you spend starting out. We haven't even begun advertising yet and some do crowdfund ( Dappervolk, FlightRising) but we're going to try to avoid that till Alpha gets much closer.
People without the ability to draw their own pet designs and items will have a bit more of a overhead because they need to outsource and spend money to buy premades or hire artists to make sets for them of items and pets for there, that's also not including world map, coding, interface/implementing features. You want to make sure you secure people in these roles who will be steady or are potentially on board long term because if they do all the skeletal structure and leave, adding a new person into something they didn't start can be a pretty big headache.
The best thing you can do is reach out to people offering the service or who have the background and cutting to the chase. What is their hourly rate, for artist what is there per pet or per item cost, do they do bulk sets, ectera. Some may even offer discounts.
We cut out a huuuuuge chunk though by having two artists, ourselves, to make the basics. However we also have had some revamps over the years. We'll use the starting creature I designed waaaay back when I first noodled I would pursue this dream.

This was the first ever Sceon and originally called a Droinlis

This was the second redraw, better for edits as we plan to make items that allow users to do custom edits. (some RLC some In-game currency)

and this is what we settled on a 3rd redraw without the noseplating.
Each one of these had about 7 colorations before we settled for final design and that took a good deal of time. So if not a premade, you may see redraws and revamps needed before launch a lot more then you like depending on the style of your pet site you are going for. Ours will be style friendly and intentionally host species designed by different artist overtime for their base creation in time.