Hi Stolarski:
Welcome to the HUBB community.
Although I live in Canada, I keep a motorcycle in Europe, and Portugal is one of my favourite countries to ride in. It's a delightful place and the cost of accommodation and food is below the European average.
I can't help you with any recommendations for bike rental. But, if you plan to ride in Europe for more than 3 weeks, I suggest you consider shipping your motorcycles from Israel to Europe. I say that because 3 weeks is about the break-point for North American residents... if we are planning a European ride that will last longer than 3 weeks, it's generally cheaper for us to ship our motorcycles (by air) to Europe than it is to rent bikes in Europe.
You might not have many options for air shipping from Israel to Europe, because El Al operates narrow-body aircraft (Boeing 737s) on routes serving continental Europe, and motorcycles only fit into the cargo holds of wide-body aircraft. But, you might be able to easily stuff all your motos into a 20 foot container and ship it by sea - that would likely cost less than air shipping.
It is very easy to arrange insurance for your Israel-plated motorcycles - heck, your local insurance company might even be able to extend your coverage to include Europe.
As for the weather, May and June are perfect months. The weather will be pleasant (very temperate) up in the north, where the mountains and nice twisty roads are, and quite warm down in the south.
Personally, I like the north of the country more than the south - there's more history, more varied topography, more interesting roads to ride, and fewer tourists. But, the whole country is interesting, and it's not that big a country... you can easily move from one end to another over the course of 3 or 4 days of relaxed touring (or 1 day of hard slogging). The portion of Spain that is directly north of Portugal is also great riding - the
Picos de Europa are just a half-day's ride to the north-east of Portugal, and that whole area of Spain is spectacular.
The only other advice I would offer is to bring good quality safety gear (helmets, gloves, riding jackets & pants, etc.). The roads are twisty, and Portugal's auto accident record is pretty poor relative to the rest of Europe.
Regards, Michael