You've received some very solid tips, but you might also try concentrating on one hotel program rather than several. For example, if you focus on Hilton you should apply for:

Amex Hilton for 50K points
Citi Hilton for 50K
BoA Virgin Atlantic for 100K (after transfer)
BoA and Bank of Hawaii Hawaiian for 140K (after transfer)
Citi Hilton Reserve for two free nights
Chase Amtrak for 50K (after transfer) and another 14K year later

Or you could focus on Choice Privileges Points that are great for European redemptions and get the following:

Barclays Choice Privileges for 40K points
Amtrak for 75K (after transfer) and 21K next year

Or you could go for Hyatt and get the following:

Chase Sapphire Preferred for 40K points (after transfer)
Chase Ink Bold for 50K points (after transfer)
Chase Hyatt for 2 free nights at any Hyatt

Or, for SPG just apply for both personal and business Amex SPG and get 50K points.

You will, of course, get even more points after meeting spending requirements, and any one of those strategies would land you more points than you need for your short vacation.

Of course, you shouldn't do it at the same time, but you have plenty of time anyway.

Also, some of those cards carry annual fees and steep spending requirements, so you'll need to investigate the further, but on the other hand you will get more points after meeting them.