ATMs and currency
Do some research to find a credit/debit card with very low rates for withdrawing cash abroad. Surprisingly, there are some credit cards which offer much better rates than debit/cash cards, such as Creation.co.uk and the Halifax Clarity card in the UK. If you forget to do this, you could be throwing money away needlessly. Ideally (and I should really follow this advice myself), you should use only use this card for ATMs, so that you can easily pay off the card in full after each withdrawal. Then use a separate credit card for online transactions, preferably with some kind of rewards (like loyalty points for airlines).
It's almost always best to wait for a new country to withdraw the majority of a new currency, but it's obviously handy to have some beforehand for transport from the airport. I try to make sure I keep a small amount of cash when leaving a country and save it for when I return.
Sometimes ATM charges are unavoidable but make sure you try a few different banks to find the lowest fee. If you're comfortable with having large amounts of cash (or have somewhere secure to keep it), then it makes sense to withdraw the maximum allowed each time, as the fees are fixed regardless of the amount you take out. I usually take my money belt with me when I need to withdraw a large amount.
Most ATMs give you large notes but this can cause problems when paying for cheap items, so I recommend always having smaller notes on you - this can be done safely at convenience stores like 7-Eleven.
Only higher-end hotels, restaurants and shops will take credit cards, so you'll be using cash for almost everything.
Budgeting
Apart from a few exceptions, my accommodation is always between $4 and $10 per night. In cheaper areas I can get a nice big room to myself with A/C and in more expensive places I'll have to go for a dorm to stick to my budget. Local food is always the cheapest, most delicious and interesting way to be fed with main meals for as little as $1.
According to IndieTraveller, these are the expected backpacker budgets per day (from cheapest to most expensive):
Vietnam: $20-30
Cambodia: $20-30
Laos: $25-35
Northern Thailand: $30
Indonesia: $30-40
Myanmar: $30-40
Malaysia: $30-45
Philippines: $35-45- Southern Thailand: $50+
Singapore: $40-90
I wouldn't take too much notice of the absolute values (for instance I live comfortably with my own room and scooter in Northern Thailand for less than $15 per day) but it is probably a good comparison between regions.
Using a budgeting app on your phone to keep track of all expenditure can be a good idea if you are worried about making your money last.
Haggling & Tipping
Most street vendors and all forms of taxi drivers will give you a starting price much higher than they would for a local. This is especially true for very touristy areas where they often get away with over-charging people who have just arrived on holiday. When getting a taxi, always try to get a metered one (in Bangkok if you ask for metered, some drivers will refuse so just find another one) as it'll always be cheaper. Always agree a price first otherwise you could be in for a surprise and/or argument. Obviously each region will have different rules on haggling/bartering so it's probably worth finding out first. When it comes to things like fruit stalls without prices, I tend to wait until a local buys something and I watch to see how much they pay.
Tipping isn't part of the culture but you will find service charges in the higher end tourist restaurants.