Yes, you can get both without paying. Bubbles come mostly from playing. Free Gold comes mostly from the Earn tab in the shop's Gold section, from other players, and from rewards.
How do I earn Gold for free?
Watch an ad: in the Earn tab, watch a video ad for Gold. The amount changes from ad to ad and is shown before you watch.
Play partner games: the Earn tab lists games from partners. Download one, finish the listed tasks, and the Gold is added in Highrise.
Answer surveys: survey partners in the Earn tab pay Gold for completed surveys.
Sell on the Marketplace: list items you own and collect the Gold when they sell.
Crew Gifts: when someone in your Crew buys a pack that includes a Crew Gift, the other members each get one, which can include Gold or Bubbles.
Tips and trades: other players can tip you or send you Gold in a trade.
Rewards: events, Daily Rewards, and promo codes sometimes include Gold.
How do I earn Bubbles?
Claim Daily Rewards from the main menu or the shop's Earn tab.
Recycle items you don't need. Items you wear in your closet give Threads, and other item types, like furniture, give Bubbles.
Take part in events and finish Tasks.
Open Crew Gifts.
For more on spending them, see What are Bubbles?
Why don't I see ads or offers?
Ads and offers come from outside partners, so what's available depends on your region, your device, and the time of day. Some players see no ads at all for a while. The Earn tab and Daily Rewards are only in the iPhone, iPad, and Android apps. See Why can't I watch an ad or see an offer?
Will Support give me free Gold?
No. Support can't give out Gold on request, including for birthdays, giveaways you missed, or reaching a goal.