[SalesForce] How to get the ID of the newly created record

As per your discussion with Nick, the view from where you are seeing the records has a filter enabled on it. It may be showing recently opened records
You can see all records by creating a new view ( if not already present ) which will not have any filter and shows all records.

Regarding your question about inserting Picklist values, here is a sample code
Project__c prj = new Project__c();
prj.Project_Name__c='Proj 003';
prj.Project_Status__c = 'Active';
insert prj;

Here Status field is a picklist value. It is ultimately a string field and a string value is saved.

Hope it is clear, you can try and get back in case of any issues.

Change this:

vendorObj.Current_work_order__c = wo.Name;

To this:

vendorObj.Current_work_order__c = wo.Id;

There seem to be other problems with your code that could cause problems - I'm not sure if they are because you cut the code down or if they are real, but I'd look closely at this code:

string venId
for(vendor__c ven : vendor)
{
    venId = ven.Id;
}

This is going to save only the last vendor id - is that what you were intending?

EDIT You don't need any of the query at the end either. You need to make this a before update, before insert trigger

Ie, remove this:

Vendor__c vendorObj = [select Id, Name, current_work_order__c from vendor__c where id =: venId];
vendorObj.Current_work_order__c = wo.Name 
//update the vendor object now
update vendorObj;

Define your trigger like this:

trigger vendorTrigger on vendor__c (before insert, before update) 

Add this:

vendor.Current_work_order__c = wo.Id;

The reason for this is that before insert/update triggers implicitly update the trigger - you are modifying the trigger before it has been saved to the database and it will be saved soon after your trigger runs.