Normalizing an apartment rental database
I have added some comments to your code since, as comments have stated, the relational design is really hard to comment on since we don't know all the business requirements.

There is no need to have an order by with an insert statement since when querying, 
the order isn't deterministic unless you specify an ORDER BY. Thus, attempting to insert
data in an order isn't really necessary

insert into bldg_table (bldg)
Select distinct bldg from apartmentmaster order by bldg;

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insert into id_rec (id, firstName, lastName)
Select id, tenantfirstname, tenantlastname 
from apartmentmaster where apt not in (603, 602);

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In the first two queries you didn't fully qualify your column names as there was no reason to. In this segment, and the following ones, you used a full qualifier of table.column. Since you are selecting from a single table, this is unnecessary. If you do want to fully qualify it, I would use table aliases. See below for the change. 

insert into apt_bldg_table(aptNumber, baseRent, bldg)
select aptmstr.apt, aptmstr.baseRent, aptmstr.bldg
from apartmentmaster as aptmstr;

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Instead of repeating this for Feburary, March, etc as you stated below,
you could create a dates table variable and use CROSS JOIN to eliminate the need to run this five times. See this link for a quick demo. Also, if you want to hard code the date as you have done, and want to use double quotes
you need to SET QUOTED_IDENTIFIER OFF.

insert into tenant_apt_table (id, aptbldg, begdate, baseRent)
select apartmentmaster.id, aptbldg_table.rec_no, "2015-01-01", aptbldg_table.baseRent
from apartmentmaster
join apt_table
on apartmentmaster.apt = aptbldg_table.aptNumber and
apartmentmaster.bldg = aptbldg_table.bldg;

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insert into rent_rec (tenant_apt_id, paid_date, paid_amt)
select
    tenant_apt_table.rec_no, apartmentmaster.janPaid, apartmentmaster.janRent
from apartmentmaster
join apt_bldg_table
on apartmentmaster.Apt = apt_bldg_table.aptNumber and apartmentmaster.Bldg = apt_bldg_table.bldg
join
    tenant_apt_table
on tenant_apt_table.aptBldg = apt_bldg_table.rec_no

/*and we repeat this query five more times for February, March, etc.*
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