Learn Hardcore Croatian: Teško mogu zapamtiti sve nove riječi bez dobrog podsjetnika. - I can hardly memorize all the new words without a good reminder.

In this sentence teško is an adverb meaning with difficulty / hardly / it’s hard to.
It modifies the verb mogu (I can), so Teško mogu zapamtiti means I can (only) with difficulty remember / I can hardly remember.
The adjective form is težak (hard, heavy), but here you need the adverb because it describes how you can do something.

Why is it mogu (I can) and not ne mogu (I can’t), even though English says I can hardly remember?

In English, can hardly is grammatically negative, but Croatian usually keeps the verb positive and lets teško carry the idea of difficulty.
So Teško mogu zapamtiti… literally is I can, but with difficulty, remember…, which corresponds to English I can hardly remember….
You could say Jedva mogu zapamtiti… with a very similar meaning. Ne mogu zapamtiti… would be stronger: I can’t remember… (not just “hardly”).

Why is the verb zapamtiti in the infinitive after mogu?

The verb moći (to be able, can) is a modal verb and it is followed by an infinitive in Croatian, just like can + infinitive in English.
So you have mogu zapamtiti = I can remember.
If you conjugated zapamtiti instead (for example zapamtim), the sentence would change meaning and sound more like a general statement: Teško zapamtim sve nove riječi ≈ It’s hard for me (whenever I try) to remember all the new words (more habitual).

What is the difference between zapamtiti and pamtiti?

Zapamtiti is perfective – it focuses on successfully completing the act of remembering: to memorize, to get something into memory (and have it stay there).
Pamtiti is imperfective – it focuses on the ongoing state or process of remembering: to remember / keep in mind / be in the habit of remembering.
In this context, zapamtiti fits well because you’re talking about managing to memorize all the new words, not just having them vaguely in mind over time.

What case is riječi in, and why does it look like that?

Here riječi is accusative plural feminine, the direct object of zapamtiti (remember what? → all the new words).
The noun riječ (word) has the form riječi for:

genitive singular (of the word),
dative/locative singular (to/at the word),
nominative plural (words),
accusative plural (words).

In this sentence, the role in the sentence (direct object) tells you it’s accusative plural: zapamtiti sve nove riječi = remember all the new words.

Why is it sve nove riječi and not something like nove sve riječi?

The usual, natural order in Croatian is quantifier – adjective – noun, so:
sve (all) + nove (new) + riječi (words).

You could move them around in special contexts, but:

sve nove riječi is neutral and normal,
nove sve riječi would sound odd or poetic at best,
sve riječi nove could appear in emphatic or poetic style, but not as the neutral everyday choice.

For learners, sve nove riječi is the pattern you should stick to.

Why is it bez dobrog podsjetnika and not bez dobar podsjetnik?

The preposition bez (without) always takes the genitive case in Croatian.
So the noun podsjetnik (reminder, masculine) must be in genitive singular: podsjetnika.
The adjective dobar (good) must agree with that: masculine, singular, genitive → dobrog.
Hence: bez dobrog podsjetnika (without a good reminder).

What gender and case is podsjetnika, and what is its base form?

The base (dictionary) form is podsjetnik (masculine noun, nominative singular).
In the phrase bez dobrog podsjetnika, it’s:

masculine,
singular,
genitive, required by bez.

So the pattern is: bez + genitive → bez podsjetnika, bez dobrog podsjetnika.

Could I say Teško zapamtim sve nove riječi bez dobrog podsjetnika instead of Teško mogu zapamtiti…? What’s the difference?

Yes, you can, but the nuance changes.

Teško mogu zapamtiti…
Emphasizes ability right now: I can hardly manage to memorize them (I am able, but with difficulty).
Teško zapamtim… (present, perfective)
Sounds more like a general tendency or repeated situation: Whenever I need to memorize them, it’s hard for me.

Both are understandable; the original with mogu is closer to English I can hardly remember in focusing on ability.

Where would I put ja if I wanted to explicitly say “I” in the sentence?

The most neutral and common placement is at the start:
Ja teško mogu zapamtiti sve nove riječi bez dobrog podsjetnika.

You could also say:

Teško ja mogu zapamtiti… – this sounds more emphatic, almost like: It’s me who really can’t / can hardly do it.

In normal speech, Croatians usually omit ja because mogu already shows 1st person singular.

Can the word order change, for example to Sve nove riječi teško mogu zapamtiti bez dobrog podsjetnika?

Yes, Croatian allows relatively flexible word order, but it changes emphasis.

Teško mogu zapamtiti sve nove riječi… – neutral focus on the difficulty of remembering.
Sve nove riječi teško mogu zapamtiti… – puts strong emphasis on all the new words (as opposed to some of them).

The original is the most neutral for simply stating the idea; the reordered version sounds more contrastive or emphatic.

Is there any difference between teško and jedva in this kind of sentence?

Both can translate as hardly / barely, but the feeling is slightly different:

Teško mogu zapamtiti… – focuses on difficulty in general: it’s hard for me to memorize….
Jedva mogu zapamtiti… – stronger idea of almost not at all: I barely manage to memorize….

They overlap a lot; in everyday speech they’re often interchangeable, with jedva sounding a bit closer to almost impossible.

Does podsjetnik mean only a physical object, or can it be digital or abstract?

Podsjetnik is quite flexible. It can mean:

a physical note (a written reminder),
a phone reminder / app notification,
even something abstract that reminds you (for example, a person or a situation).

In this sentence, it just means some kind of good reminder, without specifying the form; context would clarify what exactly.