To make the point extra clear:
は marks the topic of the sentence. が marks the subject of the sentence and is often used with specific verbs, such as 分かります=To understand.
(それが分からない, not ...は分からない)
Because of this, the が would stress your 私 in your examples in a way you'd probably not desire.
私は寒いです。= I'm cold.
私が寒いです。= Its is me, not him, not her, but me, that is cold. (implication)
Also, は can be translated as 'As concerning me, ...'. You're making yourself a topic.
Imagine a situation where a teacher asks:
田中さんはどなたですか。= 'Who is Tanaka?'
Here, 私が田中です would sound more natural than 私が田中です。
The first implies that I'M Tanaka, whilst the latter presents it like a random fact.
If I were you, I'd use Minna no Nihongo. If you want something a bit more playful, then try learning from 'Let's Learn Japanese', which can be found on Youtube I believe. Good luck!