What type of glue should be used for HDPE sheet to glue to itself
And what type of glue for HDPE be glued to wood?
Using search terms "Adhesive for HDPE" returns a number of useful links.
From a commercial site (Permabond™) it is suggested to use Permabond TA4605, TA4610, TA4611, or TA4631. Of course, considering that the linked site is for a specific line of product, all the information is positively referenced.
Another not-so-commercial site, craft art provides a more general list of products as well as considerations for cleanliness and application as well as expected cure times. Those products appear to be epoxy types, two part liquids with varying strengths. As an epoxy product, one can expect the bonding to work well to wood, the second part of your question.
As with any type of adhesive, clean surfaces are critical. In the case of the wood and HDPE surfaces, one should also abrade (roughen) those locations of adhesive application.
The entire craft art page is comprehensive and useful reading.