Long story short - to survive legendary crabs with the early crabs, as a solo player, you must play defensively. Unless you are much bigger than them, with a strong weapon, or they are already low, you will rarely be able to kill them due to their much higher base stats and strong offensive abilities. So the best thing you can do is just avoid them.

Best crabs to avoid fights against legendaries, from best to worst:

• Spanner Crab (instantly teleport to a random place on the map - instant escape). This crab is the absolute best for survival (if you don't care about playing offensively), since you can almost always get out of bad situations. You can get unlucky with this, but it's very rarely a worse situation than the one you left.
  
• Giant Mud Crab (instantly flip nearby crabs). Not-so-early, super-rare tier, but it's strong and easier to get than legendaries so it's worth a mention. This ability is great because it can be used well both offensively & defensively - you can flip crabs you're attacking, or if someone is running after you, you can just flip them and run away.
  
• Hermit Crab (gain invulnerability/max health for ~20 seconds). Most of the time people will just leave you alone instead of waiting for it to end - but if they do wait for you, you'll at least recover your boost and have a better chance at running away, and even if they stick around they will usually try and fight stuff before you come out, giving you a better chance at running away.
  
• Rainbow Crab (drop goo that stop enemies in their tracks for a good 5-10 seconds). Usually they will not bother chasing you afterwards unless you're king since this ability is really annoying to deal with. However not great against the Demon Crab if he really wants to get you.
  
• Brown Crab (infinite boost for a while). Run for your life. Pretty self-explanatory.
  
• Stone Crab (form a shield that absorbs tons of damage). Not the best while small, you'll still die quick, but as you gain some size this ability becomes really powerful and can save you from most crabs you'll come across or annoy them enough to get off of you.
  
• Blue Crab (freeze enemies in place). Similar to rainbow crab but not as effective, doesn't last long enough or have enough range for fast crabs to leave you alone.

Mantis Crab's movement is also great if you happen to get that one, but I wanted to focus more on the earlier crabs. I would place it at the top just under spanner crab, for survival.

Best hat for survival: Shark hat. Speed helps you get away. If you don't have the shark hat, try to choose the best defensive hat you have. Increased boost regen or reduced damage hats are also good.

Optional of course, but it may also help to play a little bit of "Friendly" mode (PvE) with new crabs until they're level ~15-20 before hopping into "King" (PvP), so you don't start out so tiny.

Other guides cover this pretty well, so I'll just say this: if you aren't gonna pay (I don't recommend it) get the mobile app and watch ads for rewards. Collect free pearls (20 a day) and mini-crab-grabs. Save up to 220-230 pears for the legendary crab you want (they randomly come into the shop for pearls). I would advise against the robot crab, it seems to aim very inconsistently, but if it's the one you want then go for it, the base stats are still great. If you get all the free pearls, you can get a free legendary after 10 days without even playing (even faster if you do play, due to rank & season rewards). You can also reset completed quests by watching ads, allowing you to get season rewards quicker. If you aren't willing to watch ads or pay, it's gonna be a big grind - I hope you enjoy the early game crabs.

I have been playing for 26 days with 65 hours (not counting time watching ads, but that is always while doing something else anyways), and I now have 2 legendary crabs, both with probably 50% of their upgrade points. I only spent 10$ on the season pass, but that barely accelerated the process except for upgrade points and a small amount of pearls (directly buying pearls would be better if you just want to get a legendary). However, again, I don't recommend spending money because you don't get that much out of it, especially given the amount of freebies on the app, and if you can't enjoy the game with early crabs you probably won't enjoy it for long even with legendaries.

I want to end the guide by saying, at the end of the day it's a game, so play however you want if you don't want to play defensively! However you simply won't have much to defend against legendaries if you choose something like the mitten crab or elbow crab - as strong as they may seem early on. Legendaries are somewhat OP in that sense, their offensive abilities blow earlier ones out of the water, but it's to be expected with crabs' base stats scaling up with their tiers. It gives us all something to look forward to/grind for. At least you have good defensive options to get away from them.