After getting married, what is the best way to invest our savings? Should it be under one of our names or can we invest jointly?

One of the best ways to do simple estate planning for newlyweds is to have accounts in Joint Tenancy With Rights of Survivorship. This means that upon the death of one of the joint owners, the surviving joint owner immediately owns the entire account thus avoiding having to probate the asset which is known to be difficult in many states such as California, Florida, etc. 

It does, however, depend on what the purpose for the savings is. Sometimes it could make more sense for the account to be owned by one spouse but have a "Transfer on Death" designation which would also avoid probate but only one person would have access. At the death there would still need to be some efforts taken for those funds to be given to the beneficiary.