Both my girlfriend and I are earning about mid $3000 a month. Would it be a stretch to go for an EA or EC? Or should we stick to a BTO, then upgrade later?

I would recommend sticking to a BTO so that you and your wife could save up or invest to buy a new EC or EA after satisfying the 5 years MOP. This is because in the current climate where interest rates are rising, it may not be prudent to buy EC or EA as you would have to pay more for interests.

Get a bto first before jumping into an ea/ec or private property. This will help you with managing your cash flow as the properties you mentioned will require substantial mortgage and both you and your partner may not be to support it financially. Typically more grants for bto as well. Probably a good time to sit down and plan your finances together before jumping into any purchase. Hope this helps

I think is better to BTO HDB first. Make full use of all subsidies from govt and get a BTO. After that you can either sell or upgrade

I think going for BTO and taking all the grants available for the government would be a good first step.

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in the future when your salaries have grown, maybe consider other options. Who knows maybe you would go for a condo instead ?