Finding an apartment in Taiwan might be a little tricky for foreigners. Many do not speak or read Chinese, don't know the rules and the laws. In Taiwan landlords used to be very powerful and in Taipei in the old times. People just got kicked out from one day to another. But there are laws now that can prevent this from happening to you.

contracts  
Rental contracts can be bought in a bookstore and most contracts are in Chinese language.
Get someone who can help you reading it.
You'll need two copies one for you and one for the landlord.
Make sure both contracts are identical.
If something gets written into the contract make sure that it will be stamp using the Chinese signature stamp over each additional writing.
There is no last month rent in Taiwan most landlords take two month deposit.
If you don't have the money for moving in you often times can negotiate two month for moving in and then working up the second month deposit by paying one and a half month rent the following two month.

Before moving out

Either you completed the usual 1 Yeah term or most contracts allow you to give a four weeks notice on which you can get your full deposit back.

Give the apartment back three days before the month ends and make it cash for keys.

Have witnesses around and keep one last furniture in the place which you can then grab and go.

If you give back a nice clean and intact apartment and the landlord refuses to pay back the deposit you are legally still in the place. Tell him either the deposit or he can come back in three days.

If you ruined the apartment be fair and pay for the damage.

Once you give him the key get it documented that he gave you the deposit back.

I never heard of someone fighting things in cord here and it doesn't make much sense. Find another job an work some extra hours to make up for losses.

change your locks after you moved in it'll cost you about 3000 Nt$ best is you install it yourself.

So nobody else knows which key can open that door.

Don't be surprised there are extra charges you'll have to pay which aren't listed in your contract like elevator bill and the security guard hallway and stairway lights etc. Make sure you know how much it 'will be before signing the contract.

So how to find those apartments?

One way will be to look in the streets you want to live. Apartment owners will usually hang big signs into the apartments window. Write down the numbers and call them. Many of those signs are hung out by agencies who do the job for the landlord make sure you know how much commission they'll take from you.

Now last thing my own experiences except one apartment landlord.

What you see is what you get! The owner will say “we will clean we will paint we will renovate!”

forget it!

I only met one Landlord who really put in those second set of windows we asked for and a new kitchen and all the others just cared for you to pay and be trapped with them. And for those who pass a dirty pest infested death and tbc stinky place to you. Many times they'll still be registered in your place because they aren't paying tax. They'll have their fingers in your mailbox every day. And when you are finished cleaning and renovating they'll threaten you to kick you out if you don't like to share your mailbox. And my trick with those three days left is really required to get your money back.

Good luck