Hello Thow
There is no 'optimal' way to allocate savings. There is only "your way," and that depends on your own personal goals, risk tolerance, income, ta liability, astrological sign, favorite ice-cream flavor and opinion of Taylor Swift's singing abilities.
First, I'd encourage you to come up with an Investment Policy Statment (IPS). That will help you decide where you want to put money, and in what amounds.
Second, I'd ask you how dead-set you are on purchasing a home in this time frame. If you are fairly certain you want to buy a home in under 3 years, I would start saving all of that money into a simple savings account. Just to be clear, this is what *I* would do. The reason I would do this is that it is a very short time frame, and the return on investment isn't nearly as important as having that money there in a down payment when you need it (in other words: you don't want the market to drop 20% a few weeks or months before you are ready to make an offer on a home).
However, if you are flexible on your home purchasing time-line, and it could be anywhere from 2-5+ years, I'd put it all in my index fund. If the market takes a drop, you have to be ok with delaying your home purchase for a while. If it goes up (which it tends to do about 70% of years) - hey... you a larger down payment than you need! Great!
As for investing in general - since you don't have any company matching I'd recommend funding your IRA first (and an HSA if you have one), then your 401(k), and then put the rest into taxable accounts.